Virtual probing

ABSTRACT

A method and apparatus for generating one or more transfer functions for converting waveforms. The method comprises the steps of determining a system description, representative of a circuit, comprising a plurality of system components, each system component comprising at least one component characteristic, the system description further comprising at least one measurement node and at least one output node, each of the at least one measurement nodes representative of a waveform digitizing location in the circuit. One or more transfer functions are determined for converting a waveform from one or more of the at least one measurement nodes to a waveform at one or more of the at least one output nodes. The generated transfer functions are then stored in a computer readable medium.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication No. 60/835,029, filed Aug. 2, 2006, entitled METHOD ANDAPPARATUS FOR PROBE COMPENSATION, currently pending, and U.S.Provisional Patent Application No. 60/861,678, filed Nov. 29, 2006,entitled PROBE COMPENSATION, currently pending, the entire contents ofeach of these applications being incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates to probes and probing systems especiallyas it relates to their usage with digital storage oscilloscopes.

BACKGROUND OF THE INVENTION

A digital storage oscilloscope (DSO) is one of the most commonly usedinstruments by engineers and scientists to display and analyze waveformsin electrical circuits. Often, a probe is utilized to connect the DSO toa circuit being measured and provides a mechanism for transmitting thesignal in the circuit to the oscilloscope input.

There are certain important features or characteristics of probes worthnoting. The first characteristic is the probe's loading. Since the probeis connected to a circuit under test, the loading can affect the circuitor change its operation. Since the probe usually shunts the circuit, theloading is best when the probe looks like a high impedance shunt elementto the circuit. Another feature of interest is the ability of the probeto be connected at the correct measurement point and is usuallyaddressed by making the tip of a probe as small as possible and byproviding electrical contacts in the circuit that can be placed as closeas possible to the desired measurement point. Still another feature isthe impedance match of the probe output to the DSO channel input. Whenthe probe's output impedance is matched to the DSO channel input, thetransmitted signal passes into the scope undisturbed. Finally, and noless important is the general signal fidelity aspects internal to theprobe such as magnitude and phase response.

Generally, when all of the aforementioned probe characteristics aregood, the probe is useful for use with the DSO in the acquisition andmeasurement of signals occurring in the circuit under test.

As of late, high performance DSOs and probes are being utilized tomeasure extremely fast circuits and signals. The speed of these circuitsis described by the frequency and risetimes of signals present in thecircuits. When the frequencies of interest are very high and therisetimes are very short, it becomes very difficult to develop DSOs andprobes with precision and accuracy sufficient for making accuratemeasurements. Specifically, many of the characteristics previouslydescribed become degraded.

At high frequencies, a probe's loading usually gets worse, causing theprobe to interact more and more with the circuit being measured. Whenloading effects worsen, the probe sets up reflections due to impedancediscontinuities in the circuit at the probing point. High speedmeasurements are often made on very small circuit features which make itdifficult to access the desired measurement point. Often the desiredmeasurement point is inside a chip and is completely inaccessible. It isnot possible to make a perfect impedance match to the scope at highfrequencies. This is a problem with the DSO design as well as the probedesign. Finally, the probe exhibits more and more signal fidelitydegradations internally as frequencies increase.

The problem of internal probe signal fidelity degradation has beenaddressed through the use of digital compensation. These are digitalsignal processing (DSP) methods that build filters to compensate for theprobes response. The DSP methods are utilized inside the DSO usingdigital filters stored in the probe and are generated during a probecalibration step in the manufacture of the probe. Alternatively, thesefilters are derived from probe response information stored in the probeand generated during a probe calibration step in the manufacture of theprobe.

In a coarse manner, the filters or responses that are stored in theprobe have been utilized to compensate for probe loading effects.However, this compensation can only act in a coarse manner because thecompensation methods currently used address only a fixed compensationthat addresses the loading as if the probe was being utilized in theexact environment in which it was calibrated.

These filters may comprise analog or digital filters. Analogcompensators, such as those described in U.S. Pat. No. 6,856,126 issuedto McTigue, for example, comprise fixed filters, and are therefore notadjustable in accordance with the present invention. Even prior digitalcompensators fail to address the issues noted above. Typically, thistype of digital probe compensation is handled in two ways—either thescope and probe combination is calibrated together on a given channel(the probe compensation is valid only so long as the probe is connectedto the scope channel on which it is calibrated—it cannot be moved toanother channel or another scope), or the scope/probe interface isdesigned to extremely high tolerances so that it can be ignored and isnot considered and the probe compensator forms a fixed response.

US patents to Sekel U.S. Pat. No. 6,870,359 and Pupalaikis U.S. Pat. No.6,701,335 provide for dynamic self calibration which is capable ofcompensating nearly from the probe tip through the scope (there willalways be some portion of the tip not included in his calibrationmethod). These methods are capable of accounting for the separate probeparts and the scope probe interface because the calibration signal couldbe injected near the tip and measured through the entire channel.However, this type of calibration requires complex circuitry andprocedures to accomplish the compensation. While this method doesprovide a benefit of accounting for changing conditions of the partsover time and temperature as his does, the system is not one which canbe easily implemented at, for example, a remote location not having anappropriate test and calibration fixture.

What is needed is a solution for addressing the probe loading problemsin a general sense (i.e. considering the circuit under test). What isneeded is a solution for probing inaccessible points. What is needed isa solution for dealing with the mismatch between the DSO and the probe.What is needed is a solution for determining the error bounds onmeasurements utilizing probes.

As a final note, the probe is generally manufactured with separate partsthat are assembled. Often these parts are assembled by the scope andprobe user at the time of use. Mechanisms have been provided for thestorage of separate information corresponding to the individual probeparts. What is needed is a solution that combines measurements ofseparate probe portions into an aggregate probe measurement and usingthis probe measurement for effective probe compensation.

OBJECTS OF THE INVENTION

It is an object of this invention to provide a method of probecompensation that provides for accurate measurements when the probe isutilized in a wide variety of situations.

It is another object of this invention to compensate for probe loadingeffects in a general sense.

It is a further object of this invention to provide for signalmeasurements in inaccessible locations (i.e. at points where the probecannot be physically connected).

It is a further object of this invention to provide for compensation ofthe impedance mismatch between the DSO and the probe.

It is a further object of this invention to provide compensation evenwhen the individual probe parts are interchangeable.

It is a further object of this invention to provide for thedetermination of error bounds on measurements utilizing probes.

Still other objects and advantages of the invention will in part beobvious and will in part be apparent from the specification and thedrawings.

SUMMARY OF THE INVENTION

In accordance with the invention, a method and apparatus are providedthat consider the oscilloscope and probe interface in its entirety bymeasuring at least the return loss of the probe output and the scopeinput, storing these measurements in the oscilloscope and probe, andprocessing these measurements to form a better compensation. This methodand apparatus also allow for independent measurement and calibration ofeach of the probe's parts, each of these measurements later beingcombined to provide a composite measurement of the probe.

Therefore, the invention includes the capability to provide virtualprobe compensation, i.e. providing voltages that are present at anyother points in a circuit other than where the probe tips are able to beconnected. Thus, it is possible to determine a voltage value at a pointin a circuit, even if it is not possible to directly connect a probethereto. Also, this invention is capable of determining voltages thatwould be present at the location of the probe connection as well asother points in the circuit as if the probe were not connected. This isbecause even perfect calibration of a probe to a probe tip, as addressedby the prior alt noted above, cannot provide measurements of the voltagepresent at other nodes into the circuit, or as if the probe were notconnected without considering the characteristics of the circuit undertest. It is only the present invention that can provide these benefits.

Therefore, in accordance with the invention circuit and probecharacteristics may be provided, filters may be calculated and appliedto digitized waveforms to produce other waveforms representative ofother voltages in the system, especially the concepts of generatingthese waveforms for other locations in the circuit, and/or as if theprobe were not connected.

The general situation is shown in FIG. 1. In FIG. 1, a typical desire isto measure the voltage at the load. A probe is inserted into the middleof the transmission line as shown in FIG. 2. It is shown inserted intothe middle of the line to represent the situation in which the probecannot physically access the terminals at the load or to represent thatit is more advantageous to probe the system at a different location thanthe load.

When the probe is connected to the DSO, the terminal point is the loadinside the DSO.

A probe often consists of multiple parts. It typically consists of thetip, the body, and a cable from the body to the scope connection. In acalibration phase, all of these probe parts are characterized and thedata that characterized the probe parts are stored in a non-volatilestorage element inside the probe and accessible by the scope when theprobe is inserted. Similarly, in a calibration phase of the scope, thescope input is characterized and the data is stored inside the scope.

When the probe is inserted into the scope, connecting it to a scopechannel, the scope detects the probe insertion and reads out the datastored in the probe. Often this data is represented by s parameters andthe s parameters of the probe form a system represented by a signal flowgraph as shown in FIG. 4. The signal flow diagram of the probe parts arereduced to a single two-port representation. This representation alongwith the reflection coefficient of the scope input are utilized in aflow diagram as shown in FIG. 5. FIG. 5, shows the scope and probe alongwith an idealized tee used to pick off the transmission line. Thediagram without the probe connected is represented in FIG. 3.

It is becoming increasingly common that engineers utilizing themeasurement equipment have measurements or can obtain measurements ofthe s parameters of the circuit under test. These can be generated by avector network analyzer (VNA) or through simulation. In some instances,they can be estimated crudely, such as stating that the transmissionline is such a length with such a characteristic impedance. Either way,the diagram of FIG. 5 can be thought to represent the complete circuitunder test with the probe connected.

Utilizing the flow diagrams of FIG. 3 and FIG. 5, the system can besolved for a variety of voltage relationships with respect to thevoltages of any sources applied. Furthermore, the flow diagrams can besolved for a variety of voltage relationships, all of which are relativeto the voltage appearing at the scope input in FIG. 5, independent ofthe source voltage applied. Using these voltage relationships, digitalfilters are generated and combined with scopes internal filters used forcompensation of input signals to provide a substantially correct versionof the signal appearing at any of the nodes shown in FIG. 3 or FIG. 5due to a voltage digitized by the oscilloscope.

FIG. 3 and FIG. 5, along with accompanying s parameter measurements ofthe circuit elements, present an accurate circuit description to thelevel of accuracy of s parameter measurements. If s parametermeasurements are simulated or approximated, it is possible to furthercalculate the errors present in the compensation, thus placing errorbounds on the measurements, both in the frequency domain and the timedomain.

This summary provided pertains to the simple case of a single-endedsystem with a single-ended probe and one generator. It will becomeapparent in the detailed description of the many embodiments that morecomplicated systems can also be compensated.

BRIEF DESCRIPTION OF DRAWINGS

For a more complete understanding of the invention, reference is made tothe following description and accompanying drawings, in which:

FIG. 1 is a typical single-ended circuit to be measured;

FIG. 2 is a typical single-ended measurement configuration;

FIG. 3 is a signal flow graph representing the typical single-endedcircuit to be measured in FIG. 1;

FIG. 4 is a signal flow diagram containing single-ended probe parts;

FIG. 5 is a signal flow diagram representing the typical single-endedmeasurement configuration as shown in FIG. 2;

FIG. 6 is a signal flow diagram representing the circuit in FIG. 1 withthe probing point identified;

FIG. 7 is a typical differential circuit to be measured;

FIG. 8 is a typical differential measurement configuration;

FIG. 9 is a signal flow graph representing the typical single-endedcircuit to be measured in FIG. 7;

FIG. 10 is a signal flow diagram containing differential probe parts;

FIG. 11 is a signal flow diagram representing the typical differentialmeasurement configuration as shown in FIG. 8;

FIG. 12 is a signal flow diagram representing the circuit in FIG. 7 withthe probing point identified;

FIG. 13 is a signal flow graph showing a differential probingconfiguration utilizing a 6-port to represent the probe connection;

FIG. 14 is a signal flow graph showing a differential probingconfiguration utilizing two single-ended probes;

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows a typical single-ended circuit to be measured. The circuitis single-ended because all voltages of interest along the circuit arereferenced to ground. The circuit consists of a voltage source [1] inconjunction with a source impedance [2] driving a transmission line [3]into a load [4] with a given load impedance. The transmission line is atypical medium for data communications. A transmission line will have acharacteristic impedance and an electrical length. It is understood thatthe circuit is shown simplistically and that the transmission lineactually represents a much more complicated situation. It will be shownthat the simplicity of the situation represented by FIG. 1 in no waydegrades the validity of the compensation as FIG. 1 is only a graphicalmethod to visualize the situation.

In order to measure a voltage in this circuit, a probe is inserted asshown in FIG. 2. The probe is shown as a box, representing the probebody [5], connected through a line, representing the probe tip [6] anddriving a transmission line (usually a cable) [7] of its own into a loadat the scope channel input [13]. The probe tip [6] is shown as breakingthe transmission line of the circuit into two portions—a left [8] and aright [9] portion and is shown connected to the circuit in a “tee”arrangement [10]. The left [8] and right [9] portions of thetransmission line in FIG. 2 equals the entire transmission line [3] inFIG. 1. Generally, a tee would never be utilized to split a microwavecircuit because it generates an impedance discontinuity, but the probeimpedance has been purposely designed to be very high. If it wereinfinitely high, the tee in conjunction with the infinitely highimpedance degenerates to a microwave element that has no effect on thetransmission line at all. This can be seen more clearly by analysis ofsubsequent signal flow diagrams.

In FIG. 2, the probe is shown breaking the transmission line torepresent a multitude of situations. One situation is that the engineermaking a measurement intended to measure the voltage at a particularlocation along the line at the point of probe connection [10]. Othersituations that are equally likely is that the engineer intended tomeasure the voltage at the entry to the transmission line [11] (i.e. thesource) or at the load [12]. As mentioned previously, it is often notpossible to get physical access to the source or load directly. Anotherpossibility that will be shown to be enabled by this invention is thatthe engineer wants to measure the voltage at the load, but the signal isso attenuated by the transmission line that he cannot get a low noisemeasurement at this point. What the engineer would like is to measurethe voltage at a location where the signal was larger relative to thenoise and infer the voltage at a point that was more attenuated. Ameasurement at the source would generally provide such a situation andis not precluded here. If the user can get direct access to the sourceat [11], then the situation where the left portion of the transmissionline [8] is nothing and is simply a special case of that shown in FIG.2.

When the probe is inserted as in FIG. 2, the signal at the probing point[10] is transmitted to the scope input [13] and digitized by the scope,thus measuring the waveform voltage. This arrangement, however, has beenshown to have many sources of degradation of the measurement accuracy.Two already mentioned are probe loading and the location of themeasurement. Since the probe cannot present infinite impedance, itserves to load the circuit at the probing point. The probe impedancesets up an impedance discontinuity in the circuit and therefore causesreflections. Since the probe may not be positioned exactly at the loador source, the measurement may not even be measuring the voltage ofinterest, but also the effect of reflections tends to worsen the furtherone gets from the source or load point. The probe itself and thetransmission line it drives are connected to the scope making agenerally imperfect impedance match, so the scope itself sets upreflections. Usually, the probe is designed with very high reverseisolation, so these reflections usually don't present a problem to thecircuit, but they may present a problem in the measurement.

Because of all of the difficulties mentioned, circuit designersincreasingly rely on simulation and modeling. This simulation andmodeling cannot supplant actual measurements because they can only beused to the extent that the simulation or model matches reality and onlymeasurement can determine how well it matches. Some things, however, areeasier to simulate and model (and measure) than others. For example,many techniques exist to measure characteristics of circuits. Thespecific techniques especially relevant to this invention aremeasurements of linear circuit characteristics as are performedutilizing a vector network analyzer (VNA), time domain reflectometryinstrument (TDR), or time domain network analyzer (TDNA). Theseinstruments are capable of determining the behavior of multi-portsystems to various stimuli. In a sense, they are utilized to generate amodel to describe what voltages would be measured if various stimuliwere applied. What is being proposed here is a method of combining VNAor TDR measurements that determine the behavior of the system to variousstimuli with actual scope measurements determined by the system behaviorand the actual system stimuli to predict how the scope measurement isaffected and further counteract or compensate for the effects.

The invention relies on VNA or TDR measurements that produce multi-portmeasurements usually represented by scattering parameters (or sparameters) and on methods of constructing and analyzing system modelsutilizing these s parameters and methods of signal flow graph analysis.Both s parameters and signal flow graphs are well known to those skilledin the art of the engineering of microwave systems. A signal flow graphrepresentation of the system in FIG. 1 is shown in FIG. 3. The flowgraph shows a two port model of the transmission line, a reflectioncoefficient for the source and load and a driving source. The diagramhas four dependent variables, the forward and backward propagating waveat the source (sa and sb, respectively) and the forward and backwardpropagating wave at the load (la and lb, respectively). The actualvoltage measured at the source is the sum of the voltages sa and sbmultiplied by the square root of a normalizing reference impedance Z₀used in the s parameter measurements (usually 50 Ohms). The same is truefor the voltages at the load.

Nomenclature used in these signal flow diagrams is as follows. First, amulti-port device is shown as a device with circular terminals labeledwith either an “a” or a “b”, and a number. The “a” represents a waveinput to the port and a “b” represents a wave output from the port. Thenumber represents the port number. Devices are connected by abutting “a”terminals of one device with “b” terminals of the other device andvice-versa. Devices are labeled with a box in the middle indicating thename of the device. Inside the device, arrows are shown connecting theterminals. Each arrow shown internal to the device indicatesmultiplication of a wave propagating from an “a” terminal to a “b”terminal with the understanding that the wave is multiplied by an sparameter of the device. The implied s parameter that is not shown for aconnection from a port “a[x]” to port “b[y]” internal to a devicelabeled “[N]” is “s[n][y][x]”. For example, FIG. 3, the implied sparameter between the port labeled “a1” and the port labeled “b2” of thedevice transmission line labeled “T” is “st21”. When arrows are shownexternal to a device, either the multiplication factor is directlyindicated, or is assumed to be unity if not indicated. For single portdevices, “Γ” replaces “s” and the port numbers are excluded. Forexample, FIG. 3, the implied s parameter between the port labeled “a1”and the port labeled “b1” of the load labeled “L” is “γ/”. s parametersare frequency dependent parameters.

In these and subsequent descriptions, signal flow diagrams, such as FIG.3, are said to represent a circuit topology in the sense that they showcircuit elements and their connections. The points of connection arecalled nodes and are generally labeled. As mentioned, pairs of nodesform voltages (with a multiplicative constant of the square root of thereference impedance). These are called node voltages and are referencedto ground. Pairs of node voltages form differential node voltages(through subtraction) or common mode node voltages (through averaging).The associated nodes that form differential and common mode voltages areapparent through labeling. For example, a set of nodes labeled tlpa,tlpb, tlma, tlmb would form node voltages Vtlp and Vtlm (for the plusand minus voltages referenced to ground, respectively). If thedifferential voltage was the voltage of interest, these might be furthergrouped to form the voltage Vtl, where the differential nature isimplied, but will be understood from the context.

The s parameter measurements of the circuit elements describe input andoutput characteristics of the elements. The circuit topology (includingnode connections) along with the measurements of the input and outputcharacteristics of the circuit elements (including the probes and thescope channels) form a circuit model.

In accordance with the present invention, at the time of probemanufacture, the s parameters are measured for the various portions ofthe probe independently. These portions generally include the tip, thebody and the cable to the scope. These are all measured and stored in amachine readable form that is linked in some manner to the part itself.Usually, this is done by placing the s parameters in nonvolatile memoryinside the part itself. The same is done for the scope at the time ofits manufacture, specifically the reflection coefficient of the scopeinput channel. At the time that the probe is connected to the scope, thescope obtains all of these s parameters and builds conceptually a signalflow graph as shown in FIG. 4. In FIG. 4, the probe is shown as a probetip [18], labeled “I”, a probe body [19], labeled “B” and the probecable and connection to the scope [20], labeled “O”. FIG. 4 asserts thefollowing relationships: $\begin{matrix}{{\begin{bmatrix}{{spi}\quad 11} & {{spi}\quad 12} \\{{spi}\quad 21} & {{spi}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{pia} \\{pbb}\end{bmatrix}} = \begin{bmatrix}{pib} \\{pba}\end{bmatrix}} & {{Equation}\quad 1} \\{{\begin{bmatrix}{{spb}\quad 11} & {{spb}\quad 12} \\{{spb}\quad 21} & {{spb}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{pba} \\{pob}\end{bmatrix}} = \begin{bmatrix}{pbb} \\{poa}\end{bmatrix}} & {{Equation}\quad 2} \\{{\begin{bmatrix}{{spo}\quad 11} & {{spo}\quad 12} \\{{spo}\quad 21} & {{spo}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{poa} \\{scb}\end{bmatrix}} = \begin{bmatrix}{pob} \\{sca}\end{bmatrix}} & {{Equation}\quad 3}\end{matrix}$

Equation 1, Equation 2 and Equation 3 leads to the following equation:$\begin{matrix}{{\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 11} & 1 & 0 & {{- {spi}}\quad 12} & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 12} & 0 & 1 & {{- {spi}}\quad 22} & 0 & 0 & 0 & 0 \\0 & 0 & {{- {spb}}\quad 11} & 1 & 0 & {{- {spb}}\quad 12} & 0 & 0 \\0 & 0 & {{- {spb}}\quad 21} & 0 & 1 & {{- {spb}}\quad 21} & 0 & 0 \\0 & 0 & 0 & 0 & {{- {spo}}\quad 11} & 1 & 0 & {{- {spo}}\quad 12} \\0 & 0 & 0 & 0 & {{- {spo}}\quad 21} & 0 & 1 & {{- {spo}}\quad 22} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix} \cdot \begin{bmatrix}{pia} \\{pib} \\{pba} \\{pbb} \\{poa} \\{pob} \\{sca} \\{scb}\end{bmatrix}} = \begin{bmatrix}{bsi} \\0 \\0 \\0 \\0 \\0 \\0 \\{bso}\end{bmatrix}} & {{Equation}\quad 4}\end{matrix}$

The solution to Equation 4 is given by: $\begin{matrix}{{\begin{bmatrix}{pia} \\{pib} \\{pba} \\{pbb} \\{poa} \\{pob} \\{sca} \\{scb}\end{bmatrix} = {\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 11} & 1 & 0 & {{- {spi}}\quad 12} & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 12} & 0 & 1 & {{- {spi}}\quad 22} & 0 & 0 & 0 & 0 \\0 & 0 & {{- {spb}}\quad 11} & 1 & 0 & {{- {spb}}\quad 12} & 0 & 0 \\0 & 0 & {{- {spb}}\quad 21} & 0 & 1 & {{- {spb}}\quad 21} & 0 & 0 \\0 & 0 & 0 & 0 & {{- {spo}}\quad 11} & 1 & 0 & {{- {spo}}\quad 12} \\0 & 0 & 0 & 0 & {{- {spo}}\quad 21} & 0 & 1 & {{- {spo}}\quad 22} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}^{- 1} \cdot \begin{bmatrix}{bsi} \\0 \\0 \\0 \\0 \\0 \\0 \\{bso}\end{bmatrix}}}\quad} & {{Equation}\quad 5}\end{matrix}$

Equation 5 allows for the solution of an equivalent set of s parametersfor the probe given by: $\begin{matrix}{{\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} \\{{sp}\quad 21} & {{sp}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{pia} \\{scb}\end{bmatrix}} = \begin{bmatrix}{pib} \\{sca}\end{bmatrix}} & {{Equation}\quad 6}\end{matrix}$

The s parameters in Equation 6 are solved utilizing Equation 5:$\begin{matrix}{\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} \\{{sp}\quad 21} & {{sp}\quad 22}\end{bmatrix} = \left\lbrack {\begin{matrix}\underset{\_}{pib} \\{pia} \\\underset{\_}{sca} \\{pia}\end{matrix}{{\begin{matrix}\quad \\{{bso} = 0} \\\quad \\{{bso} = 0}\end{matrix}\begin{matrix}\underset{\_}{pib} \\{scb} \\\underset{\_}{sca} \\{scb}\end{matrix}}}\begin{matrix}\quad \\{{bsi} = 0} \\\quad \\{{bsi} = 0}\end{matrix}} \right\rbrack} & {{Equation}\quad 7}\end{matrix}$FIG. 5 shows the signal flow diagram representing FIG. 2. FIG. 5 assumesthat the system s parameters are provided. As mentioned, these weremeasured on the circuit. Generally, the capability will exist to measurethe entire transmission line in the circuit, but simulation andcalculation are often necessary to separate the transmission line suchthat it can be represented by two separate sets of s parametersrepresenting the left portion and right portion relative to theanticipated probing point. In other words, the left portion of thetransmission line [21] in cascade with the right portion [22] isequivalent to the entire transmission line [14]. Given the probe sparameters calculated in Equation 7 as $\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} \\{{sp}\quad 21} & {{sp}\quad 22}\end{bmatrix}$corresponding to the probe [24], the reflection coefficient of the scopeγsc calibrated and stored in the scope corresponding to the scope input[25], the reflection coefficients $\begin{bmatrix}{{stl}\quad 11} & {{stl}\quad 12} \\{{stl}\quad 21} & {{stl}\quad 22}\end{bmatrix}\quad{{and}\quad\begin{bmatrix}{{str}\quad 11} & {{str}\quad 12} \\{{str}\quad 21} & {{str}\quad 22}\end{bmatrix}}$corresponding to the left portion [21] and the right portion [22] of thetransmission line [14], the source reflection coefficient γscorresponding to the source [15], the load reflection coefficient γ/corresponding to the load [16] and the known s parameters of a tee [23]which are $\begin{bmatrix}{{- 1}\text{/}3} & {2\text{/}3} & {2\text{/}3} \\{2\text{/}3} & {{- 1}\text{/}3} & {2\text{/}3} \\{2\text{/}3} & {2\text{/}3} & {{- 1}\text{/}3}\end{bmatrix}\quad$(independent of the reference impedance), the signal flow diagram shownin FIG. 5 is constructed. The flow graph in FIG. 5 dictates thefollowing set of relationships: $\begin{matrix}{{\begin{bmatrix}{{stl}\quad 11} & {{stl}\quad 12} \\{{stl}\quad 21} & {{stl}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{sa} \\{tlb}\end{bmatrix}} = \begin{bmatrix}{sb} \\{tla}\end{bmatrix}} & {{Equation}\quad 8} \\{\quad{{\begin{bmatrix}{{str}\quad 11} & {{str}\quad 12} \\{{str}\quad 21} & {{str}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{tra} \\{lb}\end{bmatrix}} = \begin{bmatrix}{la} \\{trb}\end{bmatrix}}} & {{Equation}\quad 9} \\{{\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} \\{{sp}\quad 21} & {{sp}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{pia} \\{scb}\end{bmatrix}} = \begin{bmatrix}{pib} \\{sca}\end{bmatrix}} & {{Equation}\quad 10} \\{{\begin{bmatrix}{{- 1}/3} & {2/3} & {2/3} \\{2/3} & {{- 1}/3} & {2/3} \\{2/3} & {2/3} & {{- 1}/3}\end{bmatrix} \cdot \begin{bmatrix}{tla} \\{trb} \\{pib}\end{bmatrix}} = \begin{bmatrix}{tlb} \\{tra} \\{pia}\end{bmatrix}} & {{Equation}\quad 11} \\{{{bs} + {\Gamma\quad{s \cdot {sb}}}} = {sa}} & {{Equation}\quad 12}\end{matrix}$

Equation 8, Equation 9, Equation 10, Equation 11 and Equation 12 lead towing equation: $\begin{matrix}{{\begin{bmatrix}1 & {{- \Gamma}\quad s} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 11} & 1 & 0 & {{- {stl}}\quad 12} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 21} & 0 & 1 & {{- {stl}}\quad 22} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & {1\text{/}3} & 1 & 0 & {{- 2}\text{/}3} & 0 & 0 & 0 & {{- 2}\text{/}3} & 0 & 0 \\0 & 0 & {{- 2}\text{/}3} & 0 & 1 & {1\text{/}3} & 0 & 0 & 0 & {{- 2}\text{/}3} & 0 & 0 \\0 & 0 & 0 & 0 & {{- {str}}\quad 11} & 1 & 0 & {{- {str}}\quad 12} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & {{- {str}}\quad 21} & 0 & 1 & {{- {str}}\quad 22} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & {{- \Gamma}\quad l} & 1 & 0 & 0 & 0 & 0 \\0 & 0 & {{- 2}\text{/}3} & 0 & 0 & {{- 2}\text{/}3} & 0 & 0 & 1 & {1\text{/}3} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {sp}}\quad 11} & 1 & 0 & {{- {sp}}\quad 12} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {sp}}\quad 21} & 0 & 1 & {{- {sp}}\quad 22} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- \Gamma}\quad{sc}} & 1\end{bmatrix} \cdot \begin{bmatrix}{sa} \\{sb} \\{tla} \\{tlb} \\{tra} \\{trb} \\{la} \\{l\quad b} \\{pia} \\{pib} \\{sca} \\{scb}\end{bmatrix}} = \begin{bmatrix}{bs} \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0\end{bmatrix}} & {{Equation}\quad 13}\end{matrix}$

Equation 13 can be solved as: $\begin{matrix}{\begin{bmatrix}1 & {{- \Gamma}\quad s} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 11} & 1 & 0 & {{- {stl}}\quad 12} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 21} & 0 & 1 & {{- {stl}}\quad 22} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & {1\text{/}3} & 1 & 0 & {{- 2}\text{/}3} & 0 & 0 & 0 & {{- 2}\text{/}3} & 0 & 0 \\0 & 0 & {{- 2}\text{/}3} & 0 & 1 & {1\text{/}3} & 0 & 0 & 0 & {{- 2}\text{/}3} & 0 & 0 \\0 & 0 & 0 & 0 & {{- {str}}\quad 11} & 1 & 0 & {{- {str}}\quad 12} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & {{- {str}}\quad 21} & 0 & 1 & {{- {str}}\quad 22} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & {{- \Gamma}\quad l} & 1 & 0 & 0 & 0 & 0 \\0 & 0 & {{- 2}\text{/}3} & 0 & 0 & {{- 2}\text{/}3} & 0 & 0 & 1 & {1\text{/}3} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {sp}}\quad 11} & 1 & 0 & {{- {sp}}\quad 12} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {sp}}\quad 21} & 0 & 1 & {{- {sp}}\quad 22} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- \Gamma}\quad{sc}} & 1\end{bmatrix}^{- 1} \cdot {\quad{\begin{bmatrix}{bs} \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0\end{bmatrix} = \begin{bmatrix}{sa} \\{sb} \\{tla} \\{tlb} \\{tra} \\{trb} \\{la} \\{l\quad b} \\{l\quad{pia}} \\{pib} \\{sca} \\{scb}\end{bmatrix}}}} & {{Equation}\quad 14}\end{matrix}$

Equation 14 provides the solution for all of the waves propagating inthe circuit with respect to bs which is defined as: $\begin{matrix}{{bs} = \frac{{Vg} \cdot Z_{0}}{{Zs} + Z_{0}}} & {{Equation}\quad 15}\end{matrix}$

Zs (the source impedance [2]) is related to the source reflectioncoefficient as: $\begin{matrix}{{\Gamma\quad s} = \frac{{Zs} - Z_{0}}{{Zs} + Z_{0}}} & {{Equation}\quad 16}\end{matrix}$

bs is dependent, therefore, on the voltage Vg of the voltage source [1]which is unknown.

Equation 17 shows the relationship of the waves determined by Equation14 to the actual voltages present in the circuit: $\begin{matrix}{{\begin{bmatrix}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\end{bmatrix} \cdot \sqrt{Z_{0}} \cdot \begin{bmatrix}{sa} \\{sb} \\{tla} \\{tlb} \\{tra} \\{trb} \\{la} \\{l\quad b} \\{pia} \\{pib} \\{sca} \\{scb}\end{bmatrix}} = {\quad\begin{bmatrix}{Vs} \\{Vtl} \\{Vtr} \\{Vl} \\{Vpi} \\{Vsc}\end{bmatrix}}} & {{Equation}\quad 17}\end{matrix}$

Equation 17, therefore, provides the voltages at all nodes of thecircuit as a function of bs or using Equation 15, as a function of Vg.In accordance with the present method, the goal is to determine thevoltage relationships not the actual voltages themselves—specifically,the relationship of all of the voltages present in the circuit withrespect to the voltage present at the scope input. All relationships aregenerated by solving Equation 17 for an arbitrary (non-zero) bs andsimply dividing by Vsc: $\begin{matrix}{\begin{bmatrix}{Hs} \\{Htl} \\{Htr} \\{Hl} \\{Hpi} \\{Hsc}\end{bmatrix} = {\begin{bmatrix}{Vs} \\{Vtl} \\{Vtr} \\{Vl} \\{Vpi} \\{Vsc}\end{bmatrix} \cdot \frac{1}{Vsc}}} & {{Equation}\quad 18}\end{matrix}$

The transfer functions solved in Equation 9 show the functions thatconvert the voltage measured at the scope input to the voltage measuredat the desired node.

The transfer functions provided by Equation 18 are useful forcalculating the voltage actually present at a circuit node due thevoltage measured by the scope when the probe is connected to thecircuit. In many cases, it is desirable to infer the voltage present ata node due to the voltage measured by the scope were the probe notconnected to the circuit. This is calculated by considering the flowdiagram in FIG. 6. It is clear that the voltages present at each nodeare: $\begin{matrix}\begin{matrix}{\begin{bmatrix}1 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 1\end{bmatrix} \cdot \sqrt{Z_{\quad 0}} \cdot} \\{\begin{bmatrix}1 & {{- \Gamma}\quad s} & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 11} & 1 & 0 & {{- {stl}}\quad 12} & 0 & 0 \\{{- {stl}}\quad 21} & 0 & 1 & {{- {stl}}\quad 22} & 0 & 0 \\0 & 0 & {{- {str}}\quad 11} & 1 & 0 & {{- {str}}\quad 12} \\0 & 0 & {{- {str}}\quad 21} & 0 & 1 & {{- {str}}\quad 22} \\0 & {{- \Gamma}\quad s} & 0 & 0 & 0 & 1\end{bmatrix}^{- 1} \cdot} \\{\begin{bmatrix}{bs} \\0 \\0 \\0 \\0 \\0\end{bmatrix} = \begin{bmatrix}{Vs}^{\quad\prime} \\{Vt}^{\quad\prime} \\{Vl}^{\quad\prime}\end{bmatrix}}\end{matrix} & {{Equation}\quad 19}\end{matrix}$

The transfer function that generates the voltages at circuit nodes withrespect the voltage at the scope channel input is given by:$\begin{matrix}{\begin{bmatrix}{Hs}^{\quad\prime} \\{Ht}^{\quad\prime} \\{Hl}^{\quad\prime}\end{bmatrix} = {\begin{bmatrix}{Vs}^{\quad\prime} \\{Vt}^{\quad\prime} \\{Vl}^{\quad\prime}\end{bmatrix} \cdot \frac{1}{Vsc}}} & {{Equation}\quad 20}\end{matrix}$

Equation 18 is utilized to generate transfer functions for the displayof waveforms that would have been present at the circuit nodes if theprobe was not connected to the circuit.

s parameters have been shown as one number for a source or load and asfour numbers for a two port device. Actually, as is well known, thisrepresentation refers to a single frequency only. In other words, eachtransfer characteristic element solved for in Equation 18 as shown is asingle complex number pertaining to a single frequency. In practice, sparameters are provided for a multitude of frequencies in order tocharacterize a device. Given s parameters for a given set offrequencies, Equation 18 and/or Equation 20 must be solved for eachfrequency.

It is advantageous to utilize the solutions to Equation 18 and Equation20 to build digital filters to be utilized by the oscilloscope totransform the digitized waveform into the waveform pertaining to a givennode. While there are many ways to do this, one method will now bedescribed in greater detail.

Given, in accordance with the present method, a transfer function vectorH (which represents an arbitrary one of the transfer functionscalculated in Equation 18 and Equation 20) consisting of a vector ofcomplex numbers (i.e. the equations were solved for a multitude offrequencies). The vector is length N+1, whereby each element in thevector H_(n), nε[0, N] represents the desired response of the filter toa frequency $f_{n} = {\frac{n}{N} \cdot {F_{e}.}}$F_(e) is the last frequency in the vector. In order to generate thisresponse vector, the s parameters provided have been measured orcalculated at all of these frequencies.

Before continuing, it is useful to point out some implications of thisvector that should be considered in order to generate a useful filter.

First, the vector assumes an equal spacing in frequency from DC toF_(e). It is not always possible to meet these conditions utilizingmeasurements from a VNA. Usually, the VNA will not generate s parametersfor DC. This is easily handled by utilizing the magnitude of the sparameter very close to DC as the DC value or by extrapolation,simulation or calculation.

Second, the selection of the frequency F_(e) should be such that itextends out to the frequencies of interest possible, either to thefrequency extent that the circuit can permit or to the frequency extentof the signals being applied.

Thirdly, if equal spacing of frequencies is not possible, interpolationwill be necessary to generate intermediate points. Interpolation is awell known technique, but care will be required in that the unwrappedphase needs to be considered in any interpolation. Modern VNAs can beconfigured to avoid this complexity.

Fourthly, the number of frequency points N+1 coupled with the endfrequency F_(e) determines the frequency resolution F_(e)/N. Thefrequency resolution determines not only the accuracy to which thefinite data points represent the system, but very importantly, the timeduration of the filter. The time duration will be the reciprocal of thefrequency resolution (i.e. N/F_(e)) and clearly must represent a timesufficient for handling all of the signal propagation times in thesystem. The time duration required will depend on the nature of thecircuit, such as the transmission line lengths involved and the degreeof impedance mismatch at various circuit points. Time durations of fiveand ten times the electrical length of the longest transmission linemight be required.

Fifthly, The vector H was calculated by dividing voltages—a voltagepresent in the circuit by a voltage present at the scope input. It isclear that if the voltage present at the scope input at a givenfrequency is zero, then H_(n) will be infinite at that point. This canoccur in pathological cases such as open or shorted transmission linestubs in the circuit, or in situations where the probe has bandwidthinsufficient to infer high frequency signals present in the circuit. Insuch a situation, F_(e) must be chosen to avoid this situation, or aprobe with sufficient bandwidth must be utilized. If, as often is thecase, the combination of the scope and probe bandwidth is insufficientto match the potential frequency content in the system, the filterresponses must be rolled off—by multiplication by a filter function, forexample. In this situation, the result of the probe compensation will besuch that it will produce a substantially correct version of thewaveform present at a circuit node within the bandwidth constraints ofthe probe and the scope. This is often not a serious limitation in thatall measurements made are substantially correct to this extent.

Finally, the transfer function of the desired filter will often requirefuture values of the input sequence. This is caused by the fact thatthat the signal arriving at the scope input is delayed by the probe andits cable and therefore signals occurring inside the circuit often occurearlier in time than the signals digitized. This is not a problem ifhandled properly and can be handled in many ways. The simplest way tohandle this is to prepare the frequency response so that it reflects theresponse to a time delayed impulse. This is done by selecting a delaytime and then remembering this delay and accounting for it in the timeplacement of the sampled and filtered waveform.

-   -   a. With the aforementioned considerations taken into account,        the vector needs to be prepared for use with an inverse discrete        Fourier transform (IDFT). This is done by treating the vector as        follows:

First, apply the delay:H_(n)→H_(n)·e^(−j·2·π·) ^(n) ^(·D)  Equation 21

Remember that this will result in a delay time of D that must beaccounted for by advancing the final, filtered waveform.

Next, define the elements iε[1,N−1], H_(N+1)= H_(N−i) . This is themirroring of the complex conjugate of the response about F_(e). SetH_(N)=|H_(N)|. The new vector is K=2. N elements long.

Next, apply a scaling factor to make the IDFT work properly:$\begin{matrix}{H_{n}->{\frac{1}{K} \cdot H_{n}}} & {{Equation}\quad 22}\end{matrix}$

Finally, Calculate the IDFT as: $\begin{matrix}\begin{matrix}{{h_{k} = {\sum\limits_{n = 0}^{K - 1}\quad{H_{n} \cdot {\mathbb{e}}^{j \cdot 2 \cdot \pi \cdot \frac{n - k}{K}}}}},} \\{k \in \left\lbrack {0,{K - 1}} \right\rbrack}\end{matrix} & {{{Equation}\quad 23} - {{Definition}\quad{of}\quad{the}\quad{IDFT}}}\end{matrix}$

This is the impulse response of a filter that can be utilized directlyat a sample rate of Fs=2 Fe: $\begin{matrix}{y_{k} = {\sum\limits_{\sigma = 0}^{K - 1}\quad{x_{k - \sigma} \cdot h_{\sigma}}}} & {{Equation}\quad 24}\end{matrix}$

Oftentimes, the filter calculated in this manner will not be at thedesired sample rate of the system. To account for this, it is convertedto the appropriate sample rate through interpolation of the time domainresponse. An easier, alternative method is to utilize the Chirp ZTransform (CZT) to make the conversion.definition  of  the  Chirp  Z  Transform $\begin{matrix}\begin{matrix}{{CZT}_{\quad m} = {\frac{1}{\quad K} \cdot {\sum\limits_{k\quad = \quad 0}^{\quad{K\quad - \quad 1}}\quad{x_{\quad k} \cdot}}}} \\{\left\lbrack {A_{0} \cdot {\mathbb{e}}^{j \cdot 2 \cdot \pi \cdot \theta_{0}} \cdot \begin{pmatrix}{\quad{w_{\quad 0} \cdot}} \\{\quad{\mathbb{e}}^{\quad{j \cdot 2 \cdot \pi \cdot \quad\phi_{\quad 0}}}}\end{pmatrix}^{m}} \right\rbrack^{- k}}\end{matrix} & {{Equation}\quad 25}\end{matrix}$

Equation 25 is utilized to generate the Z transform along an arc in theZ domain. It is used here to generate the Z transform alongequi-angularly spaced points along the unit circle (but not the spacingimplied by the discrete Fourier transform (DFT)). The goal is to createa filter of L points suitable for use at a sample rate of Fs′. Thefrequency points are defined for the CZT as M+1 points, where M=L/2,θ₀=0, A₀=1, W₀=1, and$\phi_{0} = {\frac{{Fs}^{\prime}}{Fs} \cdot {\frac{1}{L}.}}$Since the arc is not entirely arbitrary as the CZT allows, Equation 25becomes: $\begin{matrix}{{H_{m} = {\frac{1}{L} \cdot {\sum\limits_{k\quad = \quad 0}^{\quad{K\quad - \quad 1}}\quad{h_{\quad k} \cdot {\mathbb{e}}^{{{- j} \cdot 2 \cdot \pi \cdot \frac{m \cdot k}{L}}\frac{{Fs}^{\prime}}{Fs}}}}}},{m \in \left\lbrack {0,M} \right\rbrack}} & {{Equation}\quad 26}\end{matrix}$

It is illustrative to note that were Fs′=Fs and were L=K, Equation 26would become simply the DFT: $\begin{matrix}\begin{matrix}{H_{n} = {\frac{1}{K} \cdot {\sum\limits_{k = 0}^{K - 1}\quad{h_{k} \cdot}}}} \\{{\mathbb{e}}^{{- j} \cdot 2 \cdot \pi \cdot \frac{n \cdot k}{K}}}\end{matrix} & {{{Equation}\quad 27} - {{Definition}\quad{of}\quad{the}\quad{DFT}}}\end{matrix}$

The frequency of each point in the CZT is: $\begin{matrix}{{f_{m}^{\prime} = {\frac{m}{M} \cdot \frac{{Fs}^{\prime}}{2}}},} & {{Equation}\quad 28}\end{matrix}$

With the determination of the vector H from Equation 26, all thatremains is to calculate the filter by computing the inverse DFT.

First, handle the case where the desired sample rate of the final filtermay be higher than the sample rate of the current filter (i.e. Fs′>Fs).If this is the case, there will be unwanted images that must be removed:$\begin{matrix}{H_{m} = \begin{bmatrix}H_{m} & {if} & {f_{m}^{\prime} \leq \frac{Fs}{2}} \\0 & \quad & {otherwise}\end{bmatrix}} & {{Equation}\quad 29}\end{matrix}$

Next, define the elements m′ε[1,M−1], H_(M+m′)= H_(M−m′) . SetH_(M)=|H_(M)|. The new vector is L=2 M elements long.

Finally, Calculate the IDFT as: $\begin{matrix}{{h_{l}^{\prime} = {\sum\limits_{m = 0}^{L - 1}\quad{H_{m} \cdot {\mathbb{e}}^{j \cdot 2 \cdot \pi \cdot \frac{m \cdot l}{L}}}}},{l \in \left\lbrack {0,{L - 1}} \right\rbrack}} & {{Equation}\quad 30}\end{matrix}$

The resulting filter is a filter that can be utilized at the sample rateFs′ as: $\begin{matrix}{y_{k} = {\sum\limits_{\sigma = 0}^{L - 1}\quad{x_{k - \sigma} \cdot h_{\sigma}^{\prime}}}} & {{Equation}\quad 31}\end{matrix}$

The filter in Equation 31 is used to convert the digitized waveform x toa new digitized waveform y that is representative of the digitizedwaveform corresponding to the waveforms present at the nodes in FIG. 5and FIG. 6 corresponding to the voltage relationships provided inEquation 17 and Equation 19.

Up to this point, only systems where the voltages are referenced toground (i.e. single-ended systems) have been considered. A much morecomplicated, but increasingly more important situation is that ofdifferential systems and differential probes.

FIG. 7 is illustrative of a differential system. FIG. 7 is analogous toFIG. 1, but is a differential case. FIG. 7 shows two voltage sources([26] and [27]) with two source impedances ([28] and [29]) each drivinga transmission line ([31] and [32]) that arrives at two loads ([34] and[35]). Note that there is also an impedance shown between the twosources [30] and between the two loads [36]. Usually, it is the intentof such a system to develop a differential voltage as measured betweenboth driving points and to receive this difference at a receiver wherethe difference in voltage at the end of the transmission line ismeasured. Usually, it is the circuit designers' intent to drive theminus voltage with an equal and opposite voltage in time with thepositive voltage. Such a situation is said to be balanced and in such asituation, there is only a differential mode component transmitted. Incases where the voltage is not balanced, there is said to be a commonmode component of the voltage transmitted.

In some cases, the differential transmission line can be viewed as twoindependent transmission lines, but often (and in reality) they shouldbe viewed as a coupled line. FIG. 7 shows a simplified view of themultiple coupling between the lines indicated by the center transmissionline [33]—in reality the situation is more complicated that that shownsince each transmission line shown also has a different propagationtime.

The signal flow graph representation of FIG. 7 is shown in FIG. 9. FIG.9 shows the differential source as two single-ended sources ([42] and[43]), a two-port model of the source impedance [44], a four-port modelof the transmission line [45], and a two port model of the load [46].

The differential measurement situation is shown in FIG. 8. The probe isshown consisting of a pair of probe tips ([37] and [38]), a probe body[39], and a cable [40] driving a load that is the scope input channel[41]. As in the single-ended situation, the probe is shown breaking thedifferential transmission line into a left portion and a right portion,with the understanding that FIG. 8 is a simplistic graphicalrepresentation of a more complicated situation which will become clearerwhen the signal flow graphs are introduced.

In accordance with the present method, at the time of differential probemanufacture, the s parameters are measured for the various portions ofthe probe independently. These are all measured and stored in a machinereadable form that is linked in some manner to the part itself. Usually,this is done by placing the s parameters in nonvolatile memory insidethe part itself. The same is done for the scope at the time of itsmanufacture, specifically the reflection coefficient of the scope inputchannel. At the time that the probe is connected to the scope, the scopeobtains all of these s parameters and builds conceptually a signal flowgraph as shown in FIG. 10. The s parameters for the portions making upthe probe are shown as the tips [47], the body [48] and the cable andconnection to the scope [49]. Note that the probe tips [47] form afour-port device (because there are interactions between the tips) andthe probe body [48], which contains a differential amplifier, is shownas a three-port device.

The flow graph in FIG. 10 indicates the following set of relationships:$\begin{matrix}{{\begin{bmatrix}{{spi}\quad 11} & {{spi}\quad 12} & {{spi}\quad 13} & {{spi}\quad 14} \\{{spi}\quad 21} & {{spi}\quad 22} & {{spi}\quad 23} & {{spi}\quad 24} \\{{spi}\quad 31} & {{spi}\quad 32} & {{spi}\quad 33} & {{spi}\quad 34} \\{{spi}\quad 41} & {{spi}\quad 42} & {{spi}\quad 43} & {{spi}\quad 44}\end{bmatrix} \cdot \begin{bmatrix}{pipa} \\{pbpb} \\{pima} \\{pbmb}\end{bmatrix}} = \begin{bmatrix}{pipb} \\{pbpa} \\{pimb} \\{pbma}\end{bmatrix}} & {{Equation}\quad 32} \\{{\begin{bmatrix}{{spb}\quad 11} & {{spb}\quad 12} & {{spb}\quad 13} \\{{spb}\quad 21} & {{spb}\quad 22} & {{spb}\quad 23} \\{{spb}\quad 31} & {{spb}\quad 32} & {{spb}\quad 33}\end{bmatrix} \cdot \begin{bmatrix}{pbpa} \\{pbma} \\{pob}\end{bmatrix}} = \begin{bmatrix}{pbpb} \\{pbmb} \\{poa}\end{bmatrix}} & {{Equation}\quad 33} \\{{\begin{bmatrix}{{spo}\quad 11} & {{spo}\quad 12} \\{{spo}\quad 21} & {{spo}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{poa} \\{scb}\end{bmatrix}} = \begin{bmatrix}{pob} \\{sca}\end{bmatrix}} & {{Equation}\quad 34}\end{matrix}$

Equation 32, Equation 33, and Equation 34 lead to the followingequation: $\begin{matrix}{{\left\lbrack \quad\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 11} & 1 & 0 & {{- {spi}}\quad 12} & {{- {spi}}\quad 13} & 0 & 0 & {{- {spi}}\quad 14} & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 21} & 0 & 1 & {{- {spi}}\quad 22} & {{- {spi}}\quad 23} & 0 & 0 & {{- {spi}}\quad 24} & 0 & 0 & 0 & 0 \\0 & 0 & {{- {spb}}\quad 11} & 1 & 0 & 0 & {{- {spb}}\quad 12} & 0 & 0 & {{- {spb}}\quad 13} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 31} & 0 & 0 & {{- {spi}}\quad 32} & {{- {spi}}\quad 33} & 1 & 0 & {{- {spi}}\quad 34} & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 41} & 0 & 0 & {{- {spi}}\quad 42} & {{- {spi}}\quad 43} & 0 & 1 & {{- {spi}}\quad 44} & 0 & 0 & 0 & 0 \\0 & 0 & {{- {spb}}\quad 21} & 0 & 0 & 0 & {{- {spb}}\quad 22} & 1 & 0 & {{- {spb}}\quad 23} & 0 & 0 \\0 & 0 & {{- {spb}}\quad 31} & 0 & 0 & 0 & {{- {spb}}\quad 21} & 0 & 1 & {{- {spb}}\quad 33} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {spo}}\quad 11} & 1 & 0 & {{- {spo}}\quad 12} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {spo}}\quad 21} & 0 & 1 & {{- {spo}}\quad 22} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{matrix} \right\rbrack \cdot \begin{bmatrix}{pipa} \\{pipb} \\{pbpa} \\{pbpb} \\{pima} \\{pimb} \\{pbma} \\{pbmb} \\{poa} \\{pob} \\{sca} \\{scb}\end{bmatrix}} = \quad\left\lbrack \quad\begin{matrix}{bspi} \\0 \\0 \\0 \\{bsmi} \\0 \\0 \\0 \\0 \\0 \\0 \\{bso}\end{matrix} \right\rbrack} & {{Equation}\quad 35}\end{matrix}$

The solution to Equation 35 is given by: $\begin{matrix}{\begin{bmatrix}{pipa} \\{pipb} \\{pbpa} \\{pbpb} \\{pima} \\{pimb} \\{pbma} \\{pbmb} \\{poa} \\{pob} \\{sca} \\{scb}\end{bmatrix} = {\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 11} & 1 & 0 & {{- {spi}}\quad 12} & {{- {spi}}\quad 13} & 0 & 0 & {{- {spi}}\quad 14} & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 21} & 0 & 1 & {{- {spi}}\quad 22} & {{- {spi}}\quad 23} & 0 & 0 & {{- {spi}}\quad 24} & 0 & 0 & 0 & 0 \\0 & 0 & {{- {spb}}\quad 11} & 1 & 0 & 0 & {{- {spb}}\quad 12} & 0 & 0 & {{- {spb}}\quad 13} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 31} & 0 & 0 & {{- {spi}}\quad 32} & {{- {spi}}\quad 33} & 1 & 0 & {{- {spi}}\quad 34} & 0 & 0 & 0 & 0 \\{{- {spi}}\quad 41} & 0 & 0 & {{- {spi}}\quad 42} & {{- {spi}}\quad 43} & 0 & 1 & {{- {spi}}\quad 44} & 0 & 0 & 0 & 0 \\0 & 0 & {{- {spb}}\quad 21} & 0 & 0 & 0 & {{- {spb}}\quad 22} & 1 & 0 & {{- {spb}}\quad 23} & 0 & 0 \\0 & 0 & {{- {spb}}\quad 31} & 0 & 0 & 0 & {{- {spb}}\quad 21} & 0 & 1 & {{- {spb}}\quad 33} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {spo}}\quad 11} & 1 & 0 & {{- {spo}}\quad 12} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {spo}}\quad 21} & 0 & 1 & {{- {spo}}\quad 22} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}^{- 1}.\begin{bmatrix}{bspi} \\0 \\0 \\0 \\{bsmi} \\0 \\0 \\0 \\0 \\0 \\0 \\{bso}\end{bmatrix}}} & {{Equation}\quad 36}\end{matrix}$

Equation 36 allows for the solution of an equivalent set of s parametersfor the probe given by: $\begin{matrix}{{\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} & {{sp}\quad 13} \\{{sp}\quad 21} & {{sp}\quad 22} & {{sp}\quad 23} \\{{sp}\quad 31} & {{sp}\quad 32} & {{sp}\quad 33}\end{bmatrix} \cdot \begin{bmatrix}{pipa} \\{pima} \\{scb}\end{bmatrix}} = \begin{bmatrix}{pipb} \\{pimb} \\{sca}\end{bmatrix}} & {{Equation}\quad 37}\end{matrix}$

The s parameters in Equation 37 are solved utilizing Equation 36:$\begin{matrix}{\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} & {{sp}\quad 13} \\{{sp}\quad 21} & {{sp}\quad 22} & {{sp}\quad 23} \\{{sp}\quad 31} & {{sp}\quad 32} & {{sp}\quad 33}\end{bmatrix} = \left\lbrack \begin{matrix}{\frac{pipb}{pipa}|_{{bso} = 0}^{{bsmi} = 0}} & {\frac{pipb}{pima}|_{{bso} = 0}^{{bspi} = 0}} & {\frac{pipb}{scb}|_{{bsmi} = 0}^{{bspi} = 0}} \\{\frac{pimb}{pipa}|_{{bso} = 0}^{{bsmi} = 0}} & {\frac{pimb}{pima}|_{{bso} = 0}^{{bspi} = 0}} & {\frac{pimb}{scb}|_{{bsmi} = 0}^{{bspi} = 0}} \\{\frac{sca}{pipa}|_{{bso} = 0}^{{bsmi} = 0}} & {\frac{sca}{pima}|_{{bso} = 0}^{{bspi} = 0}} & {\frac{sca}{scb}|_{{bsmi} = 0}^{{bspi} = 0}}\end{matrix}\quad \right\rbrack} & {{Equation}\quad 38}\end{matrix}$

When a measurement is to be performed, s parameter measurements of thecircuit are provided. Given the probe s parameters corresponding to theprobe [54] calculated in Equation 38 as $\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} & {{sp}\quad 13} \\{{sp}\quad 21} & {{sp}\quad 22} & {{sp}\quad 23} \\{{sp}\quad 31} & {{sp}\quad 32} & {{sp}\quad 33}\end{bmatrix},$the reflection coefficient Γsc that represents the scope input [55], thes parameters $\begin{bmatrix}{{stl}\quad 11} & {{stl}\quad 12} & {{stl}\quad 13} & {{stl}\quad 14} \\{{stl}\quad 21} & {{stl}\quad 22} & {{stl}\quad 23} & {{stl}\quad 24} \\{{stl}\quad 31} & {{stl}\quad 32} & {{stl}\quad 33} & {{stl}\quad 34} \\{{stl}\quad 41} & {{stl}\quad 42} & {{stl}\quad 43} & {{stl}\quad 44}\end{bmatrix}\quad$and $\begin{bmatrix}{{str}\quad 11} & {{str}\quad 12} & {{str}\quad 13} & {{str}\quad 14} \\{{str}\quad 21} & {{str}\quad 22} & {{str}\quad 23} & {{str}\quad 24} \\{{str}\quad 31} & {{str}\quad 32} & {{str}\quad 33} & {{str}\quad 34} \\{{str}\quad 41} & {{str}\quad 42} & {{str}\quad 43} & {{str}\quad 44}\end{bmatrix}\quad$that represent the left portion [50] and right portion [51] of thetransmission line [45], the s parameters $\begin{bmatrix}{{ss}\quad 11} & {{ss}\quad 12} \\{{ss}\quad 21} & {{ss}\quad 22}\end{bmatrix}\quad$that represent the source impedance [44], the s parameters$\begin{bmatrix}{s/11} & {s/12} \\{s/21} & {s/22}\end{bmatrix}\quad$that represent the load [46] and the known s parameters of a tee thatrepresent the positive probe tip connection point [52] and the negativeprobe tip connection point [53], the signal flow diagram shown in FIG.11 is constructed. FIG. 11 shows the signal flow diagram representingFIG. 8. Unlike FIG. 8, which is a simplified graphical view of thesituation, FIG. 11 is not simplified inasmuch as it considers four portmodels of the transmission line and two port models load and sourceimpedance without any other simplifying assumptions.

The flow graph in FIG. 11 dictates the following set of relationships:$\begin{matrix}{{{\begin{bmatrix}{{stl}\quad 11} & {{stl}\quad 12} & {{stl}\quad 13} & {{stl}\quad 14} \\{{stl}\quad 21} & {{stl}\quad 22} & {{stl}\quad 23} & {{stl}\quad 24} \\{{stl}\quad 31} & {{stl}\quad 32} & {{stl}\quad 33} & {{stl}\quad 34} \\{{stl}\quad 41} & {{stl}\quad 42} & {{stl}\quad 43} & {{stl}\quad 44}\end{bmatrix}\quad} \cdot \begin{bmatrix}{spa} \\{tlpb} \\{sma} \\{tlmb}\end{bmatrix}} = \begin{bmatrix}{spb} \\{tlpa} \\{smb} \\{tlma}\end{bmatrix}} & {{Equation}\quad 39} \\{{{\begin{bmatrix}{{str}\quad 11} & {{str}\quad 12} & {{str}\quad 13} & {{str}\quad 14} \\{{str}\quad 21} & {{str}\quad 22} & {{str}\quad 23} & {{str}\quad 24} \\{{str}\quad 31} & {{str}\quad 32} & {{str}\quad 33} & {{str}\quad 34} \\{{str}\quad 41} & {{str}\quad 42} & {{str}\quad 43} & {{str}\quad 44}\end{bmatrix}\quad} \cdot \begin{bmatrix}{trpa} \\{lpb} \\{trma} \\{lmb}\end{bmatrix}} = \begin{bmatrix}{trpb} \\{lpa} \\{trmb} \\{lma}\end{bmatrix}} & {{Equation}\quad 40} \\{{\begin{bmatrix}{{sp}\quad 11} & {{sp}\quad 12} & {{sp}\quad 13} \\{{sp}\quad 21} & {{sp}\quad 22} & {{sp}\quad 23} \\{{sp}\quad 31} & {{sp}\quad 32} & {{sp}\quad 33}\end{bmatrix} \cdot \begin{bmatrix}{pipa} \\{pima} \\{scb}\end{bmatrix}} = \begin{bmatrix}{pipb} \\{pimb} \\{sca}\end{bmatrix}} & {{Equation}\quad 41} \\{{\begin{bmatrix}{{- 1}/3} & {2/3} & {2/3} \\{2/3} & {{- 1}/3} & {2/3} \\{2/3} & {2/3} & {{- 1}/3}\end{bmatrix} \cdot \begin{bmatrix}{tlpa} \\{trpb} \\{pipb}\end{bmatrix}} = \begin{bmatrix}{tipb} \\{trpa} \\{pipa}\end{bmatrix}} & {{Equation}\quad 42} \\{{\begin{bmatrix}{{- 1}/3} & {2/3} & {2/3} \\{2/3} & {{- 1}/3} & {2/3} \\{2/3} & {2/3} & {{- 1}/3}\end{bmatrix} \cdot \begin{bmatrix}{tlma} \\{trmb} \\{pimb}\end{bmatrix}} = \begin{bmatrix}{tlmb} \\{trma} \\{pima}\end{bmatrix}} & {{Equation}\quad 43} \\{{\begin{bmatrix}{s/11} & {s/12} \\{s/21} & {s/22}\end{bmatrix} \cdot \begin{bmatrix}{lpa} \\{lma}\end{bmatrix}} = \begin{bmatrix}{lpb} \\{lmb}\end{bmatrix}} & {{Equation}\quad 44} \\{{{\begin{bmatrix}{{ss}\quad 11} & {{ss}\quad 12} \\{{ss}\quad 21} & {{ss}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{spb} \\{smb}\end{bmatrix}} + \begin{bmatrix}{bsp} \\{bsm}\end{bmatrix}} = \begin{bmatrix}{spa} \\{sma}\end{bmatrix}} & {{Equation}\quad 45} \\{{{{sca} \cdot \Gamma}\quad{sc}} = {scb}} & {{Equation}\quad 46}\end{matrix}$

Equation 39 through Equation 46 leads to the following equations:$s = {\begin{bmatrix}1 & {{- {ss}}\quad 11} & 0 & 0 & 0 & {{- {ss}}\quad 12} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 11} & 1 & 0 & {{- {stl}}\quad 12} & {{- {stl}}\quad 13} & 0 & 0 & {{- {stl}}\quad 14} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 21} & 0 & 1 & {{- {stl}}\quad 22} & {{- {stl}}\quad 23} & 0 & 0 & {{- {stl}}\quad 24} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & {1/3} & 1 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 \\0 & {{- {ss}}\quad 21} & 0 & 0 & 1 & {{- {ss}}\quad 22} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 31} & 0 & 0 & {{- {stl}}\quad 32} & {{- {stl}}\quad 33} & 1 & 0 & {{- {stl}}\quad 34} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {stl}}\quad 41} & 0 & 0 & {{- {stl}}\quad 42} & {{- {stl}}\quad 43} & 0 & 1 & {{- {stl}}\quad 44} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & {1/3} & 1 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 \\0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 0 & 1 & {1/3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {str}}\quad 11} & 1 & 0 & {{- {str}}\quad 12} & {{- {str}}\quad 13} & 0 & 0 & {{- {str}}\quad 14} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {str}}\quad 21} & 0 & 1 & {{- {str}}\quad 22} & {{- {str}}\quad 23} & 0 & 0 & {{- {str}}\quad 24} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- s}/11} & 1 & 0 & 0 & {{- s}/12} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 0 & 1 & {1/3} & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {str}}\quad 31} & 0 & 0 & {{- {str}}\quad 32} & {{- {str}}\quad 33} & 1 & 0 & {{- {str}}\quad 34} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 0 & {{- {str}}\quad 41} & 0 & 0 & {{- {str}}\quad 42} & {{- {str}}\quad 43} & 0 & 1 & {{- {str}}\quad 44} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- s}/21} & 0 & 0 & 0 & {{- s}/22} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & {1/3} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {sp}}\quad 11} & 1 & {{- {sp}}\quad 12} & 0 & 0 & {{- {sp}}\quad 13} \\0 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 0 & 0 & {{- 2}/3} & 0 & 0 & 0 & 0 & 1 & {{- 1}/3} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {sp}}\quad 21} & 0 & {{- {sp}}\quad 22} & 1 & 0 & {{- {sp}}\quad 23} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- {sp}}\quad 31} & 0 & {{- {sp}}\quad 32} & 0 & 1 & {{- {sp}}\quad 33} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- \Gamma}\quad{sc}} & 1\end{bmatrix}{\quad\quad}{Equation}\quad 47}$ $\begin{matrix}{{ab} = \begin{bmatrix}{spa} \\{spb} \\{tlpa} \\{tlpb} \\{sma} \\{smb} \\{tlma} \\{tlmb} \\{trpa} \\{trpb} \\{lpa} \\{lpb} \\{trma} \\{trmb} \\{lma} \\{lmb} \\{pipa} \\{pipb} \\{pima} \\{pimb} \\{sca} \\{scb}\end{bmatrix}} & {{Equation}\quad 48}\end{matrix}$ $\begin{matrix}{{bs} = \begin{bmatrix}{bsp} \\0 \\0 \\0 \\{bsm} \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0\end{bmatrix}} & {{Equation}\quad 49}\end{matrix}$ $\begin{matrix}{V = \begin{bmatrix}{Vs} \\{Vtl} \\{Vtr} \\{Vl} \\{Vpi} \\{Vsc}\end{bmatrix}} & {{Equation}\quad 50}\end{matrix}$ $\begin{matrix}{{vd} = \begin{bmatrix}1 & 1 & 0 & 0 & {- 1} & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 1 & 0 & 0 & {- 1} & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & {- 1} & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & {- 1} & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & {- 1} & {- 1} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\end{bmatrix}} & {{Equation}\quad 51}\end{matrix}$ $\begin{matrix}{V = {{vd} \cdot \sqrt{Z_{0}} \cdot S^{- 1} \cdot {bs}}} & {{Equation}\quad 52}\end{matrix}$ $\begin{matrix}{H = {V \cdot \frac{1}{Vsc}}} & {{Equation}\quad 53}\end{matrix}$

Similarly, the signal flow diagram in FIG. 12, which represents theunconnected probe case, dictates the following set of relationships:$\begin{matrix}{{\begin{bmatrix}{{stl}\quad 11} & {{stl}\quad 12} & {{stl}\quad 13} & {{stl}\quad 14} \\{{stl}\quad 21} & {{stl}\quad 22} & {{stl}\quad 23} & {{stl}\quad 24} \\{{stl}\quad 31} & {{stl}\quad 32} & {{stl}\quad 33} & {{stl}\quad 34} \\{{stl}\quad 41} & {{stl}\quad 42} & {{stl}\quad 43} & {{stl}\quad 44}\end{bmatrix} \cdot \begin{bmatrix}{spa} \\{tpb} \\{sma} \\{tmb}\end{bmatrix}} = \begin{bmatrix}{spb} \\{tpa} \\{smb} \\{tma}\end{bmatrix}} & {{Equation}\quad 54} \\{{\begin{bmatrix}{{str}\quad 11} & {{str}\quad 12} & {{str}\quad 13} & {{str}\quad 14} \\{{str}\quad 21} & {{str}\quad 22} & {{str}\quad 23} & {{str}\quad 24} \\{{str}\quad 31} & {{str}\quad 32} & {{str}\quad 33} & {{str}\quad 34} \\{{str}\quad 41} & {{str}\quad 42} & {{str}\quad 43} & {{str}\quad 44}\end{bmatrix} \cdot \begin{bmatrix}{tpa} \\{lpb} \\{tma} \\{lmb}\end{bmatrix}} = \begin{bmatrix}{tpb} \\{lpa} \\{tmb} \\{lma}\end{bmatrix}} & {{Equation}\quad 55} \\{{\begin{bmatrix}{s/11} & {s/12} \\{s/21} & {s/22}\end{bmatrix} \cdot \begin{bmatrix}{lpa} \\{lma}\end{bmatrix}} = \begin{bmatrix}{lpb} \\{lmb}\end{bmatrix}} & {{Equation}\quad 56} \\{{{\begin{bmatrix}{{ss}\quad 11} & {{ss}\quad 12} \\{{ss}\quad 21} & {{ss}\quad 22}\end{bmatrix} \cdot \begin{bmatrix}{spa} \\{smb}\end{bmatrix}} + \begin{bmatrix}{bsp} \\{bsm}\end{bmatrix}} = \begin{bmatrix}{spa} \\{sma}\end{bmatrix}} & {{Equation}\quad 57}\end{matrix}$

Equation 54 through Equation 57 lead to the following equations:$\begin{matrix}{S^{\prime} = \begin{bmatrix}1 & {{- {ss}}\quad 11} & 0 & {{- {ss}}\quad 12} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {st}}\quad 11} & 1 & {{- {st}}\quad 13} & 0 & 0 & {{- {st}}\quad 12} & 0 & {{- {st}}\quad 14} & 0 & 0 & 0 & 0 \\0 & {{- {ss}}\quad 21} & 1 & {{- {ss}}\quad 22} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{{- {st}}\quad 31} & 0 & {{- {st}}\quad 33} & 1 & 0 & {{- {st}}\quad 32} & 0 & {{- {st}}\quad 31} & 0 & 0 & 0 & 0 \\{{- {st}}\quad 21} & 0 & {{- {st}}\quad 23} & 0 & 1 & {{- {st}}\quad 22} & 0 & {{- {st}}\quad 21} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & {{- {str}}\quad 11} & 1 & {{- {str}}\quad 13} & 0 & 0 & {{- {str}}\quad 12} & 0 & {{- {str}}\quad 14} \\{{- {st}}\quad 41} & 0 & {{- {st}}\quad 43} & 0 & 0 & {{- {st}}\quad 42} & 1 & {{- {st}}\quad 44} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & {{- {str}}\quad 31} & 0 & {{- {str}}\quad 33} & 1 & 0 & {{- {str}}\quad 32} & 0 & {{- {str}}\quad 34} \\0 & 0 & 0 & 0 & {{- {str}}\quad 21} & 0 & {{- {str}}\quad 23} & 0 & 1 & {{- {str}}\quad 22} & 0 & {{- {str}}\quad 24} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- s}/11} & 1 & {{- s}/12} & 0 \\0 & 0 & 0 & 0 & {{- {str}}\quad 41} & 0 & {{- {str}}\quad 43} & 0 & 0 & {{- {str}}\quad 42} & 1 & {{- {str}}\quad 44} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{- s}/21} & 0 & {{- s}/22} & 1\end{bmatrix}} & {{Equation}\quad 58}\end{matrix}$ $\begin{matrix}{{ab}^{\prime} = \begin{bmatrix}{spa} \\{spb} \\{sma} \\{smb} \\{tpa} \\{tpb} \\{tma} \\{tmb} \\{lpa} \\{lpb} \\{lma} \\{lmb}\end{bmatrix}} & {{Equation}\quad 59} \\{{bs}^{\prime} = \begin{bmatrix}{bsp} \\0 \\{bsm} \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0 \\0\end{bmatrix}} & {{Equation}\quad 60} \\{V^{\prime} = \begin{bmatrix}{Vs}^{\prime} \\{Vt}^{\prime} \\{Vl}^{\prime}\end{bmatrix}} & {{Equation}\quad 61}\end{matrix}$ $\begin{matrix}{{vd}^{\prime} = \begin{bmatrix}1 & 1 & {- 1} & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 1 & {- 1} & {- 1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & {- 1} & {- 1}\end{bmatrix}} & {{Equation}\quad 62}\end{matrix}$ $\begin{matrix}{V^{\prime} = {{vd}^{\prime} \cdot \sqrt{Z_{0}} \cdot S^{\prime - 1} \cdot {bs}^{\prime}}} & {{Equation}\quad 63} \\{H^{\prime} = {V^{\prime} \cdot \frac{1}{Vsc}}} & {{Equation}\quad 64}\end{matrix}$

In order to generate the probe compensation filters, Equation 52 andEquation 63 are solved for the voltages present in the circuit due tobsp and bsm that make up bs (in Equation 49) and bs′ (in Equation 60)and then the transfer function for all filters are calculated usingEquation 53 and Equation 64.

The differential case has some particular details that must be handledspecially. It is clear that a differential probe measures the voltagedifference in a circuit, but the actual voltages present and indicatedby the signal flow graph are in actuality referenced to ground. In otherwords, the differential voltage is the difference between two voltagesthat are in actuality referenced to ground. It is often useful to thinkof the voltages present in the circuit not as two single ended voltages,but instead as a voltage consisting of a differential and common-modecomponent. The intent of the probe is to measure the differentialcomponent and to reject the common-mode component. It is usually thecircuit designer's intent to transmit a differential voltage and to nottransmit any common-mode components, but unintentional common modecomponents will occur. In the solutions of the single-ended case, I usedan arbitrary, non-zero value bs that was suitable for solving for thevoltage relationships in the circuit and probe, but in the differentialcase, this is no longer possible since there are two different valuesbsm and bsp that contribute to the voltages present in the circuit.While either bsm or bsp can be chosen arbitrarily (non-zero), therelationship between them is not arbitrary and must be specified. Thereare a few ways to go about determining this relationship.

The first way is when the differential probe is being utilized in asingle-ended measurement situation. This situation occurs when adifferential probe is utilized with either of its two tips connected toground. In this situation, its intent is to truly measure the voltage toground. This situation is dealt with by simply setting bsm=0 or bsp=0,depending on which tip is connected to ground. Note that in thissituation, the s parameters for the system would reflect the singleground referenced transmission line under test.

The second way is to assume balanced drive. In most cases, this is agood approximation of how the system is driven. In this case, setbsm=−bsp. The compensation filters will be valid only to the extent thatthe balanced drive condition is met. Its validity, however, will notdepend on common mode components set up in the system due to modeconversion within the circuit under balanced drive conditions, becausethese are properly accounted for in the flow graph and equations.

Finally, the degree of balance can be measured. This is accomplished bytaking measurements of the source driving a known load. This could bedone, for example, by simply driving a differential step from the sourceand acquiring the plus and minus source step on an oscilloscope. Atransfer function H_(bsp→bsm)(S) is thereby calculated that converts bspto bsm. During the solution of the aforementioned equations, anarbitrary (non-zero) value would be chosen for bsp and bsm is chosen as:bsm=bsp·H _(bsp→bsm)(j·2·π·f)  Equation 65

Equation 65 accounts for balance of frequency components, but would notaccount for changes in source balance over time.

As mentioned previously, the equations presented give solutions for asingle frequency. All frequencies are solved and the resulting frequencyresponse is converted to a filter for implementation as previouslydiscussed.

Many probe compensation situations have thus been described. Thesingle-ended measurement with a single-ended probe, the differentialmeasurement with a differential probe and the single-ended measurementwith the differential probe. Many other variations can utilize thesesame techniques described, but it is important to specifically point outtwo others.

The first is shown in FIG. 13. FIG. 13 is a variation on FIG. 11 in thata six-port model [56] replaces the two tees [52] and [53]. The intenthere is that the exact six port measurement of the probe connection tothe circuit can be measured or calculated in order to supplant the teeusage, were this found to be inadequate. It is advantageous to use amodel such as [56] when s parameter measurements can be made of theprobe in the exact environment in which it used in a circuit. As such,this method treats the probe and probe connection in the measurement asopposed to previously described methods which treat the probe tips asthe interface.

The second is shown in FIG. 14. This is a case of utilizing two singleended probes [57] and [59] applied to two scope inputs [58] and [60].This flow diagram is solved in a manner as already demonstrated toobtain values for the voltages in the system as in Equation 52. It isobvious from the previous examples how to do this. Examining howEquation 52 comes about reveals that the equation can always be reducedto the form of: $\begin{matrix}{V = {A \cdot \begin{bmatrix}{bsp} \\{bsm}\end{bmatrix}}} & {{Equation}\quad 66}\end{matrix}$This is done by simply removing the columns of vd·√{square root over(Z₀)}·S⁻¹ corresponding to the columns in bs that are zero. A istherefore a matrix with two columns and a number of rows correspondingto the rows in V. In this manner, a single voltage in the system can bewritten as: $\begin{matrix}{v = {\begin{bmatrix}a_{p} & a_{m}\end{bmatrix} \cdot \begin{bmatrix}{bsp} \\{bsm}\end{bmatrix}}} & {{Equation}\quad 67}\end{matrix}$

The voltages at the two scope inputs can also be written as:$\begin{matrix}{\begin{bmatrix}{Vscp} \\{Vscm}\end{bmatrix} = {\begin{bmatrix}b_{pp} & b_{pm} \\b_{m\quad p} & b_{m\quad m}\end{bmatrix} \cdot \begin{bmatrix}{bsp} \\{bsm}\end{bmatrix}}} & {{Equation}\quad 68}\end{matrix}$

Therefore, the transfer function of the filter would be determined bythe following equations: $\begin{matrix}{{H \cdot \begin{bmatrix}{Vscp} \\{Vscm}\end{bmatrix}} = v} & {{Equation}\quad 69}\end{matrix}$

Or: $\begin{matrix}{{H \cdot \begin{bmatrix}b_{pp} & b_{pm} \\b_{m\quad p} & b_{m\quad m}\end{bmatrix} \cdot \begin{bmatrix}{bsp} \\{bsm}\end{bmatrix}} = {\begin{bmatrix}a_{p} & a_{m}\end{bmatrix} \cdot \begin{bmatrix}{bsp} \\{bsm}\end{bmatrix}}} & {{Equation}\quad 70}\end{matrix}$

And therefore: $\begin{matrix}{H = {{\begin{bmatrix}a_{p} & a_{m}\end{bmatrix} \cdot \begin{bmatrix}b_{pp} & b_{pm} \\b_{m\quad p} & b_{m\quad m}\end{bmatrix}^{- 1}} = \begin{bmatrix}H_{p} & H_{m}\end{bmatrix}}} & {{Equation}\quad 71}\end{matrix}$

Equation 69 and Equation 71 imply correctly that in this situation, thedesired waveform is a function of the waveform acquired on each of thescope inputs. Specifically, the desired waveform is the sum of twofiltered waveforms acquired by the two scope channels. In thissituation, unlike the single differential probe situations, noassumptions are required that relate bsp and bsm. As such, theconfiguration in FIG. 14 is suitable for determination of positive,negative, differential, common-mode etc. voltage components withoutassumptions made regarding the balance of the source.

Many probing configurations and compensation methods have been provided.All of these methods assume the knowledge of the s parameters thatdefine the system. It is clear that the accuracy of these compensationmethods depend on the accuracy of the s parameters provided, but itshould also be clear that the probe compensation is beneficial even whenless than perfect statements can be made regarding the system sparameters. For example, if the circuit designer knows that he has atransmission line with certain characteristic impedance, certain losscharacteristics, and certain lengths, he can still make use of thisalgorithm by providing s parameters of the circuit element models.Similarly, assumptions of matched loads, matched sources etc. will stilloften lead to probe compensations that result in improved accuracy overpreviously used methods.

Furthermore, the methods set forth here provide for analysis of errorsthat might be introduced with respect to errors in the models. Forexample, users can often provide error bounds on circuit elementmeasurements either as peak errors or standard deviations. Similarly,scope and probe manufacturers know similar error bounds on the probe andscope parameter measurements. Using the methods set forth, these errorbounds can be utilized in, for example, a Monte-Carlo analysis, toproduce compensated waveforms that allow analysis of the effects oferrors on the time domain waveforms produced.

As many possible circuit topologies have been calculated, the method ofcalculating the probe compensation filters for arbitrary configurationsand topologies should now be understood. A MathCad spreadsheetconsisting of a four page calculation is shown in FIG. 15 through FIG.18 is provided to show how to handle arbitrary configurations andtopologies. Once a topology is known, the equations are filled in andsolved using the helper functions provided to generate the filtertransfer characteristics.

Up until this point, many various circuit topologies have been shown.Now, a method of handling completely arbitrary circuit topologies isprovided.

Consider that, based on the topologies exemplified, a generic systemdefinition consists of a text file definition that defines a picture ofa system. The system definition preferably provides all of theinformation that defines:

The devices in a system, where a device is defined by a name, a numberof ports, and the device characteristics (i.e. a set of s parameters).

The nodes in a system, where a node is defined as a connection betweenports of devices.

The stimulus in a system, where a stimulus is defined as a name and adevice port from which the stimulus emanates.

The measurement nodes, defined as signals that will be acquired.

The output nodes, defined as signals that will be produced.

A stimulus definition that will match the degrees of freedom in thesystem defined by the number of stimuli to the degrees of freedomdefined by the number of measurement nodes.

From this generic system definition, the goal is to generate filtersthat, when applied to waveforms actually acquired at the measurementnodes into waveforms that appear at the output nodes.

It is assumed here that a picture of system (that contains all of theinformation above) is equivalent to a text file system description thatcontains the same information. The linkage between the text file systemdescription and the picture is traditionally thought of as being one ofnet list generation. In other words, computer aided design (CAD) toolsthat generate pictorial circuit and system definitions convert to anequivalent text file definition through generation of a net list. Thegeneration of a net list from the picture is a trivial matter. Thisembodiment will consider the conversion of a system description in textfile form.

Given a system description file, the preferred steps in parsing thisfile are:

Count the device declarations and node declarations.

Dimension an array of devices descriptions (Devices) to the number ofdevices counted. A device description consists of a name and a set of sparameters (the name and s parameters for device d is accessed asDevices[d].Name, Devices[d].S), and is an array of ports (port p ofdevice d is accessed as Devices[d][p]). A port consists of both an A andB node connection along with a stimulus connection (these are accessedfor port p of device d as Devices[d][p].A, Devices[d][p].B, andDevices[d][p].Stim).

For each device declaration, dimension device d to the number of ports Pdeclared for the device (i.e. Devices[d] becomes a P element array ofports).

For each device declaration, assign the s parameters to the device (i.e.assign Devices[d].S). The s parameters come from a file (i.e. Touchstoneformat) or from some other specification. The s parameters areinterpolated to match the desired frequency resolution.

For each node declaration, take the node name (or create a unique nodename) and add a “1” and a “2” creating two node names. Since the nodedeclaration declares the node (named NodeName) connection from a device(named FromName) port (numbered FromPort) to another device (namedToName) port (numbered ToPort), the assignment is:

a. Devices[index(FromName, Devices)][FromPort−1].A=NodeName+“1”

b. Devices[index(FromName, Devices)][FromPort−1].B=NodeName+“2”

c. Devices[index(ToName, Devices)][ToPort].A=NodeName+“2”

d. Devices[index(ToName, Devices)][ToPort].B=NodeName+“1”

Where index(name, array) is a function that determines the index d suchthat array[d].Name=name.

For each stimulus declaration, that provides a name (Name), a devicename (DeviceName) and a port number (PortNum), assignDevices[index(DeviceName, Devices)][Portnum−1].Stim=Name. Also, placethe stimulus name in a vector of names M′.

Place each name declared as a measurement node in a vector of names VM.

Place each name declared as an output node in a vector of names VO.

Convert the stimulus definition declaration by creating a vector ofnames D of the free stimuli and a matrix R that defines the relationshipof D to M′ (i.e. R D=M′).

At this point, the information contained in Devices contains a completedefinition of the system, its connections, and its behavior. VM, VOcontain the names of input and output nodes. D, R and M′ contain therelationships of stimuli in the system.

The next step is to create vectors containing the stimuli names andvoltage node names. This is done by first determining the total numberof ports (ports) as the sum of all of the device ports in the system.$\left( {{i.e.\quad{ports}} = {\sum\limits_{d = 0}^{{{Devices} \cdot {Elements}} - 1}{{{Devices}\lbrack d\rbrack} \cdot {Elements}}}} \right).$By visiting each device, and each port in each device in order, thevoltage node names vector N and the stimuli name vector M is filled in:$\begin{matrix}{\left. i\leftarrow 0 \right.{{{for}\quad d} \in {{0\quad\ldots\quad{{Devices}.{Elements}}} - 1}}\left\{ \begin{matrix}{{{for}\quad p}\quad \in \quad{{0\quad\ldots\quad{{{Devices}\quad\lbrack d\rbrack}\quad.\quad{Elements}}}\quad - \quad 1}} \\\left\{ \begin{matrix}\left. {N\lbrack i\rbrack}\quad\leftarrow\quad{{{{Devices}\quad\lbrack d\rbrack}\lbrack p\rbrack}\quad.\quad B} \right. \\\left. {M\lbrack i\rbrack}\quad\leftarrow\quad{{{{Devices}\quad\lbrack d\rbrack}\lbrack p\rbrack}\quad.\quad{Stim}} \right. \\\left. i\quad\leftarrow\quad{i\quad + \quad 1} \right.\end{matrix} \right.\end{matrix} \right.} & {{Equation}\quad 72}\end{matrix}$

Note that the number of elements in N and M is Ports. If there is nostimulus for a device port, the element in M is empty. M now contains aPorts element vector whose elements are either empty, or contain thename of a stimulus applied to the system. N now contains a Ports elementvector whose elements contain all of the nodes defined in the system.Each element of N must be defined and have a unique name.

The next step in converting such a system description into filters isto, for each frequency of interest, generate a matrix S_(inv). This isperformed as follows, for each frequency of interest n, nε[0,N] (i.e. Nis the last frequency point of interest) $\begin{matrix}{{{{for}\quad n} \in {0\quad\ldots\quad N}}\left\{ \begin{matrix}{{{for}\quad d} \in {{0\quad\ldots\quad{{Devices}.{Elements}}} - 1}} \\\left\{ \begin{matrix}{{{for}\quad p} \in {{0\quad\ldots\quad{{{Devices}\quad\lbrack d\rbrack}.{Elements}}} - 1}} \\\left\{ \begin{matrix}\left. r\leftarrow{{index}\quad\left( {{{{{Devices}\quad\lbrack d\rbrack}\lbrack p\rbrack}.B},N} \right)} \right. \\\left. {{S\lbrack r\rbrack}\lbrack r\rbrack}\leftarrow 1.0 \right. \\{{{for}\quad c} \in {{0\quad\ldots\quad{{{Devices}\quad\lbrack d\rbrack}.{Elements}}} - 1}} \\\left\{ {{S\lbrack r\rbrack}\left\lbrack {{index}\quad\left( {{{{{Devices}\quad\lbrack d\rbrack}\lbrack c\rbrack}.A},N} \right\rbrack}\leftarrow{- {{{Devices}\quad\lbrack d\rbrack}.{{{S\lbrack n\rbrack}\lbrack p\rbrack}\lbrack c\rbrack}}} \right.} \right.\end{matrix} \right.\end{matrix} \right. \\\left. {S_{inv}\lbrack n\rbrack}\leftarrow S^{- 1} \right.\end{matrix} \right.} & {{Equation}\quad 73}\end{matrix}$

Note that in Equation 73, since S accesses the s parameters for adevice, S[n][p][c] accesses the element at row p, column c of the sparameter matrix corresponding to frequency element n.

At this point, S_(inv) is an N+1 element vector of matrices where ateach frequency point n, the following equation is obeyed:S _(inv) [n]·M=N  Equation 74

By removing the rows of M that are empty, and reordering M, one arrivesat M′. Similarly, when the columns of S_(inv)[n] corresponding to theempty rows of Mare removed and the columns are reordered in the samemanner as the rows of M were to arrive at M′, S′_(inv)[n] is producedsuch that:S _(inv) ′[n]·M′=N  Equation 75

Thus if Stims is the number of stimuli, the number of elements in M′,and recalling that Ports is the total number of device ports in thesystem, then S_(inv)′[n] is a Ports x Stims matrix. N is a Ports elementvector.

A vector V is formed containing all of the names of the voltage nodesdeclared previously. The number of nodes (Nodes), the number elements inV, is half of Ports. All of the names in V are found in N with either a“1” or “2” appended to the name. It is well known in dealing with sparameters that the incident power wave on an interface is defined interms of the voltage (v), current (i), and real reference impedance (Z0)as: $\begin{matrix}{a = \frac{v + {{i \cdot Z}\quad 0}}{2\sqrt{Z\quad 0}}} & {{Equation}\quad 76}\end{matrix}$

Similarly, the reflected power wave from an interface is defined as:$\begin{matrix}{b = \frac{v - {{i \cdot Z}\quad 0}}{2\sqrt{Z\quad 0}}} & {{Equation}\quad 77}\end{matrix}$

Therefore, the voltage is defined by the power waves as: $\begin{matrix}{v = \frac{a + b}{\sqrt{Z\quad 0}}} & {{Equation}\quad 78}\end{matrix}$

Since N contains the power waves, a voltage extraction matrix thatconverts N into voltage vector V is created such that:VE·N=V  Equation 79

VE is a Nodes x Ports element matrix and is created simply by putting a1 in each row and column where V[row] with either a “1” or “2” appendedis found in N[column]. The resulting VE is divided by √{square root over(2)} (although this falls out eventually, anyway). Substituting Equation79 into Equation 75:VE·S _(inv) ′[n]·M′=V  Equation 80

Remember, we already have the following relationship:R·D=M′  Equation 81

D has a number elements which represent the degrees of freedom in thesystem (Degrees) and R is a Stims x Degrees element matrix.

Substituting Equation 81 into Equation 80:VE·S _(inv) ′[n]·R·D=V  Equation 82

Define a new matrix:VS[n]=VE·S _(inv) ′[n]·R  Equation 83

VS is a Nodes x Degrees element matrix. Substituting Equation 83 intoEquation 82:VS[n]·D=V  Equation 84

We have already defined two vectors VM and VO that contain the names ofthe measurement and output nodes, respectively. VM has a length equal tothe number of measurement nodes (Meas) and VO has a length equal to thenumber of output nodes (Outputs). Note that both of these vectorscontain names found in V, allowing two matrices VSM and VSO to bedefined such that:VSM[n]·D=VM  Equation 85VSO[n]·D=VO  Equation 86

VSM is created by placing each row of VS corresponding to an element ofV found in VM at the row number corresponding to the element in VM. Inother words for each row rvm in VM, we find the corresponding row rv,such that V[rv]=VM[rvm] and we fill in the row rvm of VSM with the rowrv of VS. VSO is created in a similar manner using VO.

VSM is a Meas x Degrees element matrix.

VSO is a Outputs x Degrees element matrix.

If Meas=Degrees, we have a properly constrained system and we can solveEquation 85:D=VSM[n] ⁻¹ ·VM  Equation 87

Otherwise, if Meas>Degrees, we have an over constrained system, and wecan solve Equation 85, in a least mean squared error sense, as follows:D=(VSM[n] ^(T) ·VSM[n]) ⁻¹ ·VSM[n] ^(T) ·VM  Equation 88

In the properly constrained case, we substitute Equation 87 intoEquation 86:VSO[n]·VSM[n] ⁻¹ ·VM=VO

Therefore, a frequency response of a filter can be defined that convertsthe measured node voltage VM into the desired output voltage VO isdefined:H[n]=VSO[n]·VSM[n] ⁻¹  Equation 89

In the over constrained case, the frequency response becomes:H[n]=VSO[n]·( VSM[n] ^(T) ·VSM[n]) ⁻¹ ·VSM[n] ^(T)  Equation 90

Thus, regardless of whether the system is properly or over constrained,either Equation 89 or Equation 90 defines the frequency response of asystem such that:H[n]·VM=VO  Equation 91

This calculation is repeated for each frequency point n until a completefilter frequency response has been built up. Thus if m is an index intothe names of measurement nodes such that VM[m] is the name of themeasurement node m, and o is an index into the names of output nodessuch that VO[o] is the name of an output node, then H[n][o][m] is acomplex value representing the response of a filter to be applied tomeasurement node m to produce output node o at frequency point n. Theactual filter output to produce output node o at frequency point n is:$\begin{matrix}{{V\quad{O\lbrack o\rbrack}} = {\sum\limits_{m = 0}^{{Meas} - 1}{{{{{H\lbrack n\rbrack}\lbrack o\rbrack}\lbrack m\rbrack} \cdot V}\quad{M\lbrack m\rbrack}}}} & {{Equation}\quad 92}\end{matrix}$

This response generated from Equation 89 or Equation 90 is eitherutilized using frequency domain filtering methods, or is converted totime domain impulse responses using the inverse FFT or other well knownmethods of finite impulse response (FIR) or infinite impulse response(IIR) filter design methods.

It should be noted that if the number of measurement nodes is unity,then the filter will have a topology that all skilled in the art ofdigital signal processing are familiar with—a filter that produces asingle output waveform from a single input waveform. If the number ofmeasurement nodes is greater than unity, however, the topology of thefilter that produces any one output waveform from the input waveforms isone whereby a filter is applied to each of the measured waveforms andthe result of each filter is summed to form the output waveform.

A few practical considerations are worth noting. First, the number offrequency points and their locations should be chosen judiciously tofavor various techniques for conversion to time domain filters. In manycases, it is advantageous to interpolate the s parameters onto a gainfulfrequency scale.

Another consideration involves the invertibility of the matrices inEquation 89 and Equation 90. There will be problems at frequencylocations where an output voltage is desired at a frequency where nofrequency content arrives at a measurement node. This can be handledthrough bandwidth limiting, or restricting the frequency response range,in these cases. Similarly, even if the matrices are invertible, the gainof the filters can become too high resulting in unacceptableperformance. The comment on limiting frequencies of operation stillapplies in this case, as well.

Equation 91 demonstrates that filters can be generated that convertmeasured voltages in a system into output voltages. It also shows thatif output voltages and measured voltages can be acquired simultaneouslywith a variety of measured voltages, it is possible to work backwards tomeasure the desired filter responses. This is done by creating a numberof measurement cases at least equal to the number of measurement nodes(Cases≧Meas), and for each case, measuring the voltages at themeasurement nodes and the output nodes at a given frequency. In otherwords for a measurement case c, cε0 . . . Cases−1, then VMC[n][m][c] isassigned with the voltage measured for frequency n at measurement node mfor case c. Similarly, VOC[n][o][c] is assigned with the voltagemeasured simultaneously for frequency n at output node o for case c.Then, for Cases=Meas, the filter response can be calculated as:H[n]=VOC[n]·VMC[n] ⁻¹  Equation 93

In the situation where Cases>Meas, then the filter response can becalculated as:H[n]=VOC[n]·VMC[n] ^(T)·(VMC[n]·VMC[n] ^(T))⁻¹  Equation 94

It is obvious that the filter response can be calculated only if VMC[n](or VMC[n]·VMC[n]^(T)) is invertible, meaning that the measurement casesgenerate orthogonal sets of equations. In a physical sense, when thematrix can't be inverted, it means one of several possibilities that areeasily dealt with. It can mean that an output does not depend on one ofthe input voltages, meaning that the system of equations can be reducedby excluding the offending measurement node. Or it can mean that novoltage can be generated at a given frequency at one of the measurementnodes (a common situation in systems with standing waves, where nullsoccur at various locations, or in systems with serial data signals wherethere will be nulls in the frequency content), but this means, again,that either the measurement node or the frequency point can be excluded(if a frequency point is excluded, the response is set to some arbitraryvalue (usually zero) that does not affect things anyway because novoltage can be generated at that frequency. In this case, the filter maybe fully useful only for a particular serial data transmission rate.

Based on this description of a preferred embodiment, many benefits ofthese methods should be readily apparent and worth pointing out.

These methods are capable of compensating for effects of probing andmeasurement in a system, including the removal of probe loading effectsand the compensation for non-ideal probe and measurement instrumentresponse. These methods can be utilized to produce waveforms, based onacquired or simulated waveforms, that show waveforms present insituations where the measurement equipment is not connected to thecircuit. These methods can be utilized to show waveforms at circuitlocations other than the locations probed (again, under circumstanceswith and without the measurement equipment connected) and at physicallyinaccessible locations. These methods can be utilized to compensateprobes in situations where differential probes are utilized in asingle-ended measurement configuration (i.e. one tip is connected toground) and in situations where multiple single-ended probes areutilized to make differential measurements. These methods can beutilized to measure waveforms at circuits and circuit locations that arecompletely virtual, meaning that, provided with an accurate systemdescription, these methods can produce output waveforms in circuits andcircuit locations that are not physically present during themeasurement. The demonstrated and disclosed capability of this method tohandle virtually arbitrary measurement and circuit considerationsanticipates a wide variety of application.

1. A method for generating one or more transfer functions for convertingwaveforms, comprising the steps of: determining a system description,representative of a circuit, comprising a plurality of systemcomponents, each system component comprising at least one componentcharacteristic, the system description further comprising at least onemeasurement node and at least one output node, each of the at least onemeasurement nodes representative of a waveform digitizing location inthe circuit; determining one or more transfer functions for converting awaveform from one or more of the at least one measurement nodes to awaveform at one or more of the at least one output nodes; and storingthe generated transfer functions in a computer readable medium.
 2. Themethod of claim 1, whereby a transfer function of the one or moretransfer functions is adapted to produce a waveform that is asubstantially correct representation of a waveform that would be presentat the location in the circuit represented by the at least one outputnode in accordance with at least one or more waveforms that would bepresent at the one or more measurement nodes.
 3. The method of claim 2,whereby the at least one output node is representative of a location inthe circuit where a probe tip is attached thereto.
 4. The method ofclaim 2, wherein at least one of the system components is representativeof a probe attached to the circuit.
 5. The method of claim 4, wherein atleast one of the system components is representative of a measurementapparatus to which the probe is attached.
 6. The method of claim 4,wherein the at least one output node is representative of a location inthe circuit other than where the probe tip is connected.
 7. The methodof claim 4, wherein the at least one output node is representative of alocation in a corresponding circuit equivalent to the circuit but as ifthe probe were not attached.
 8. A method for generating a waveform,comprising the steps of: determining a system description in accordancewith a circuit under test comprising a plurality of system components,each system component comprising at least one component characteristic;determining one or more transfer functions for converting a waveform ata measurement node at one location in the circuit under test to awaveform at an output node of the circuit under test; receiving awaveform; and generating a waveform at the output node in accordancewith the one or more transfer functions corresponding to therelationship between the one or more measurement nodes and the one ormore output nodes.
 9. The method of claim 8, whereby the one or moretransfer functions are adapted to produce a waveform that is asubstantially correct representation of a waveform that is present atthe one or more output nodes in accordance with a waveform at the one ormore measurement nodes.
 10. The method of claim 9, whereby the outputnode is located at a portion of the circuit where a probe tip isattached thereto.
 11. The method of claim 10, wherein the probe is asingle ended probe.
 12. The method of claim 10, wherein the probe is adifferential probe.
 13. The method of claim 12, wherein one portionthereof is connected to a ground.
 14. The method of claim 13, whereinthe probe is in a single ended measurement configuration.
 15. The methodof claim 8, wherein at least one of the system components is a probeattached to the circuit.
 16. The method of claim 15, wherein at leastone of the system components is a measurement apparatus to which theprobe is attached.
 17. The method of claim 15, wherein the output nodeis representative of a location in the circuit where the probe is notconnected.
 18. The method of claim 15, wherein the output node isrepresentative of a location in a corresponding circuit equivalent tothe circuit, but as if the probe were not attached.
 19. The method ofclaim 8, wherein one or more transfer functions are determined forconverting a waveform received at a plurality of measurement nodes at acorresponding plurality of locations in the circuit under test to awaveform at an output node of the circuit under test.
 20. The method ofclaim 19, wherein the one or more transfer functions are determined forconverting a waveform received at the plurality of measurement nodes ata corresponding plurality of locations in the circuit under test to aplurality of waveforms at a plurality of output nodes of the circuitunder test.
 21. The method of claim 8, wherein one or more transferfunctions are determined for converting a waveform received at themeasurement node in the circuit under test to a plurality of waveformsat a corresponding plurality of output nodes of the circuit under test.22. The method of claim 21, wherein the output nodes are located atportions of the circuit where probe tips are attached thereto.
 23. Themethod of claim 22, wherein two of the probe tips comprise adifferential probe.
 24. The method of claim 8, wherein the systemdescription is defined in accordance with one or more s parameters foreach of the system components.
 25. The method of claim 24, furthercomprising the step of combining s parameters indicative of systemcomponents in a circuit relating the measurement node and the outputnode to generate the transfer function.
 26. The method of claim 8,wherein the output node is at an arbitrary location in the circuit undertest.
 27. The method of claim 8, wherein the output node is located at aportion of the circuit other than where the probe tip is connected. 28.The method of claim 8, wherein the output node is location at a portionof a corresponding circuit equivalent to the circuit but as if the probewere not attached.
 29. The method of claim 8, wherein the output node islocated inside an integrated circuit.
 30. The method of claim 8, whereinthe output node is located inside a closed component.
 31. An apparatusfor generating one or more transfer functions for converting waveforms,comprising: a controller, the controller adapted to: determine a systemdescription, representative of a circuit, comprising a plurality ofsystem components, each system component comprising at least onecomponent characteristic, the system description further comprising atleast one measurement node and at least one output node, each of the atleast one measurement nodes representative of a waveform digitizinglocation in the circuit; and determine one or more transfer functionsfor converting a waveform from one or more of the at least onemeasurement nodes to a waveform at one or more of the at least oneoutput nodes; and a computer readable medium for storing the generatedtransfer functions.
 32. The apparatus of claim 31, whereby a transferfunction of the one or more transfer functions is adapted to produce awaveform that is a substantially correct representation of a waveformthat would be present at the location in the circuit represented by theat least one output node in accordance with at least one or morewaveforms that would be present at the one or more measurement nodes.33. The apparatus of claim 32, whereby the at least one output node isrepresentative of a location in the circuit where a probe tip isattached thereto.
 34. The apparatus of claim 32, wherein at least one ofthe system components is representative of a probe attached to thecircuit.
 35. The apparatus of claim 34, wherein at least one of thesystem components is representative of a measurement apparatus to whichthe probe is attached.
 36. The apparatus of claim 34, wherein the atleast one output node is representative of a location in the circuitother than where the probe tip is connected.
 37. The apparatus of claim34, wherein the at least one output node is representative of a locationin a corresponding circuit equivalent to the circuit but as if the probewere not attached.
 38. An apparatus for generating a waveform,comprising: a controller, the controller adapted to: determine a systemdescription in accordance with a circuit under test comprising aplurality of system components, each system component comprising atleast one component characteristic; determine one or more transferfunctions for converting a waveform at a measurement node at onelocation in the circuit under test to a waveform at an output node ofthe circuit under test; receive a waveform; and generate a waveform atthe output node in accordance with the one or more transfer functionscorresponding to the relationship between the one or more measurementnodes and the one or more output nodes.
 39. The apparatus of claim 38,whereby the one or more transfer functions are adapted to produce awaveform that is a substantially correct representation of a waveformthat is present at the one or more output nodes in accordance with awaveform at the one or more measurement nodes.
 40. The apparatus ofclaim 39, whereby the output node is located at a portion of the circuitwhere a probe tip is attached thereto.
 41. The apparatus of claim 40,wherein the probe is a single ended probe.
 42. The apparatus of claim40, wherein the probe is a differential probe.
 43. The apparatus ofclaim 41, wherein one portion thereof is connected to a ground.
 44. Theapparatus of claim 43, wherein the probe is in a single endedmeasurement configuration.
 45. The apparatus of claim 38, wherein atleast one of the system components is a probe attached to the circuit.46. The apparatus of claim 45, wherein at least one of the systemcomponents is a measurement apparatus to which the probe is attached.47. The apparatus of claim 45, wherein the output node is representativeof a location in the circuit where the probe is not connected.
 48. Theapparatus of claim 45, wherein the output node is representative of alocation in a corresponding circuit equivalent to the circuit, but as ifthe probe were not attached.
 49. The apparatus of claim 38, wherein thecontroller determines one or more transfer functions for converting awaveform received at a plurality of measurement nodes at a correspondingplurality of locations in the circuit under test to a waveform at anoutput node of the circuit under test.
 50. The apparatus of claim 49,wherein the controller determines one or more transfer functions forconverting a waveform received at the plurality of measurement nodes ata corresponding plurality of locations in the circuit under test to aplurality of waveforms at a plurality of output nodes of the circuitunder test.
 51. The apparatus of claim 38, wherein the controllerdetermines one or more transfer functions for converting a waveformreceived at the measurement node in the circuit under test to aplurality of waveforms at a corresponding plurality of output nodes ofthe circuit under test.
 52. The apparatus of claim 51, wherein theoutput nodes are located at portions of the circuit where probe tips areattached thereto.
 53. The apparatus of claim 52, wherein two of theprobe tips comprise a differential probe.
 54. The apparatus of claim 38,wherein the system description is defined in accordance with one or mores parameters for each of the system components.
 55. The apparatus ofclaim 54, wherein the controller combines s parameters indicative ofsystem components in a circuit relating the measurement node and theoutput node to generate the transfer function.
 56. The apparatus ofclaim 38, wherein the output node is at an arbitrary location in thecircuit under test.
 57. The apparatus of claim 38, wherein the outputnode is located at a portion of the circuit other than where the probetip is connected.
 58. The apparatus of claim 38, wherein the output nodeis location at a portion of a corresponding circuit equivalent to thecircuit but as if the probe were not attached.
 59. The apparatus ofclaim 38, wherein the output node is located inside an integratedcircuit.
 60. The apparatus of claim 38, wherein the output node islocated inside a closed component.
 61. A method for generating awaveform in a circuit under test, comprising the steps of: determining asystem description in accordance with the circuit under test;determining a transfer function for converting a waveform received at ameasurement node at one location in the circuit under test to a waveformat an output node of the circuit under test; receiving a waveform at themeasurement node; and generating a waveform at the output node inaccordance with the transfer function.
 62. The method of claim 61,whereby the output node is located at a portion of the circuit where aprobe is attached thereto.
 63. The method of claim 62, wherein the probeis a single ended probe.
 64. The method of claim 62, wherein the probeis connected to a measurement apparatus.
 65. The method of claim 61,wherein the output node is representative of a location in the circuitwhere a probe is not connected.
 66. The method of claim 61, wherein theoutput node is representative of a location in a corresponding circuitequivalent to the circuit, but as if the probe were not attached.
 67. Amethod for measuring by a measurement system a waveform generated in acircuit under test, comprising the steps of: generating a systemdescription in accordance with a defined system configuration comprisinga plurality of system components, each system component comprising atleast one component characteristic; generating one or more transferfunctions for converting a waveform at one location in the measurementsystem to a waveform at another location in the measurement system;determining first system values at one or more locations in themeasurement system in response to a first input waveform; determiningsecond system values at one or more locations in the measurement systemin response to a second input waveform; and determining a resultantwaveform at one or more locations in the measurement system inaccordance with at least one of the first and at least one of the seconddetermined system values.
 68. A method for measuring by a measurementsystem a waveform generated in a circuit under test, comprising thesteps of: generating a first system description in accordance with afirst defined system configuration comprising a plurality of systemcomponents, each system component comprising at least one componentcharacteristic; generating a second system description, in accordancewith a system configuration defined by modifying one or more of theplurality of system components in the first defined systemconfiguration, comprising a plurality of system components, each systemcomponent comprising at least one component characteristic; generatingone or more transfer functions for converting a waveform at one locationin the measurement system defined by the first defined systemconfiguration to a waveform at another location in the measurementsystem defined by the first defined system configuration; generating oneor more transfer functions for converting a waveform at one location inthe measurement system defined by the second defined systemconfiguration to a waveform at another location in the measurementsystem defined by the second defined system configuration; and storingthe generated transfer functions in a computer readable medium.
 69. Amethod for probe compensation, comprising the steps of: generating afirst system component description comprising at least one componentcharacteristic; generating a second system component descriptioncomprising at least one component characteristic; generating one or moretransfer functions, in accordance with at least the first and secondcomponent descriptions, for converting a waveform at one location in themeasurement system to a waveform at another location in the measurementsystem.
 70. The method of claim 69, wherein the first system componentdescription describes an electronic probe; and wherein the second systemcomponent description describes a teat and measurement apparatus. 71.The method of claim 70, wherein the test and measurement apparatuscomprises an oscilloscope.